They did NOT tell us why 360°, but we can guess at 3 reasons, some weighted combination of which were probably important to them as to why the degree as defined was a good natural unit:

1. The year however counted (e.g., Julian year = 365.25 days exactly by definition, solar year = 365.2421897 days (J2000); sidereal year = 365.256363004 days (J2000), etc.) is ∼ 360 days. By choosing 360° in the circle, the angular velocity ω of Sun on the celestial sphere works out to be very nearly 1 degree/day. (To be exact, angular velocity ω = 0.985626 ... degree/day for the Julian year (J2000).) With a good approximate angular velocity of 1 degree/day, it is easy to calculate Sun movements approximately and mentally estimating them is easy too. So for Sun tracking purposes, the degree is a good natural unit.

   Note the Babylonian astronomers would NOT have wanted to make the degree yield exactly with in their uncertainty a solar angular velocity of 1 degree/day since that would have divided the circle into an inconvenient non-integer number of degrees. Inconvenient for mental understanding and for simple calculations including especially division.

   Why did the Babylonian astronomers want especially to keep track of the position of the Sun on the celestial sphere? Oh, astrology. If the Sun is in a particular zodiac sign at the time of your birth, then that is your zodiac sign and your whole life story can be previewed---for a fee.

2. The Babylonian astronomers (and Babylonian mathematicians too, of course) for mathematics and astronomy used a sexagesimal system (base-60 system) (which has 12 divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) and NOT a decimal system (base-10 system) which they used probably mostly in everyday life---counting on the old fingers, you know. See the Discussion of the Babylonian Sexagesimal Base System below.

   So obviously dividing the circle into a multiple of their sexagesimal base 60 would have been convenient for calculations.

   In one sense dividing the circle into 60 units would be the most consistent choice.

   But such an angular unit would be unconveniently large for precise astronomy and would need sub-units even for approximate work.

   So the obvious convenient division was into 360 units which gives units that are small enough without being too small and which give an angular velocity of close to 1 degree/day which satisfies the first reason given above.

   Also 360 is a conveniently 6 times the sexagesimal base 60. The number 6 is closely related to the sexagesimal base 60 (it's 60 divided by 10) and has 4 highly useful divisors itself: 1, 2, 3, 6.

   Note that the sexagesimal system came into use circa 2000 BCE (see Wikipedia: Babylonian numerals: Origin), and so was NOT chosen to get a good base system for Babylonian astronomy: it just worked out that way fortuitously.

   Would the Babylonian astronomers have chosen to the divide the circle into a multiple of their base 60 even if days in the year had NOT been close to any such multiple? Who knows. It might have depended on chance or on some weighting of advantages that we do NOT know of.

3. Since 360 has 24 divisors divisors (including the first 6 positive integers), division with 360 is often easy. Thus, in arithmetic with angles, dividing the circle into 360° is a great simplification since the result of many common division operations is often a whole number: i.e., a number with NO trailing digits in a decimal fraction.

   So even if the Babylonian astronomers had NOT used a sexagesimal base system, dividing the circle into 360° would have been convenient.

   See Table: The 24 Divisors of 360 below.

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   Table: The 24 Divisors of 360

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     Count       small     large
         2           1       360
         4           2       180
         6           3       120
         8           4        90
        10           5        72
        12           6        60
        14           8        45
        16           9        40
        18          10        36
        20          12        30
        22          15        24
        24          18        20 

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Discussion of the Babylonian Sexagesimal Base System:

1. The Babylonian astronomers used a sexagesimal base system (with base (AKA radix) 60) rather than a decimal base system (with base (AKA radix) 10).

   Why? Probably for easy division: 60 has 12 divisors which include the first six positive integers: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

   But we do NOT know for sure.

   But for sure it's why we have 60 seconds in a minute and 60 minutes in an hour: we got those from them, i.e., the Babylonian astronomers.

   And why 24 hours in the day? The ancient Egyptians thought it good to divide daytime and nighttime into 10 units each, but then added begining and end units to each phase to account for twilight, and so 12 units for each phase adding up to 24 units for a day. The upshot is the 24-hour day. See Ne-? or Otto Neugebauer 1969, The Exact Sciences in Antiquity, p. ?.

   Why did the ancient Egyptians start with 10 units? Oh, like most every culture in the world, they used a decimal base system some of the time because humans have 10 fingers (see below for further explication).

2. The Babylonian astronomers did use sexagesimal fractions too, but it was tricky since they didn't use a sexagesimal point---context told you where the fraction began---and if context did NOT tell, you did NOT know.
   Note whole numbers are whole numbers in any base system since they can be created by just adding a string of 1's in any base system:
   1+1+1+1+ ...

   Similarly, a fraction less than 1, remains a fraction less than 1 in any base system since 1 is 1 in any base system.

3. Note that the Babylonians used the sexagesimal base system primarily just for mathematics and astronomy. They had other base systems including the quasi-ubiquitous decimal system---quasi-ubiquitous very, very probably because humans have 10 fingers (see Wikipedia: Decimal: Decimal notation and Wikipedia: Ten).

   Other base systems turn up commonly in human societies. There are the 20-base systems (vigesimal systems) for those who like to count fingers and toes (AKA toezy woezies). The vigesimal system vestigially exists in the English word score---"Four score and seven years ago" (Gettysburg Address, 1863, Abraham Lincoln (1809--1865)).

   Then there is duodecimal system with 12 as a base (AKA radix) that turns up in some cultures. Recall the duodecimal system: 12 eggs in a dozen and 144 chickens in a gross. The duodecimal system made a lot of sense: a small enough base that the numeral digits are easily memorized, but with 6 divisors (1, 2, 3, 4, 6, 12) including the highly useful 2 3, and 4.

   The duodecimal system would have been the better choice for the conventional base system---but in the battle of fingers versus eggs, fingers won.

4. Really digressing, note that 60 and 360 are highly composite numbers. "A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer has" (Wikipedia: highly composite number). The HCNs run: (1) 1, (2) 2, (3) 4, (4) 6, (5) 12, (6) 24, (7) 36, (8) 48, (9) 60, (10) 120, (11) 180, (12) 240, (13) 360, (14) 720 (15) 840, (16) 1260, (17) 1680, (18) 2520, (19) 5040, (20) 7560, (21) 10080 ...

5. Really digressing again, note that

      1170/360 = ∑_(n=1)^(24)(1/divisor_n) 

   which is an expansion in unit fractions that are 1 over the divisors of 360. The ancient Egyptian mathematicians liked expanding their fractions in unit fractions because they seemed to think of unit fractions as more fundamental.