Because the moon is a big bright ball in the sky which regularly moves through predictable, easily observable phases, making it an obvious basis for a system of measuring time.
The complexities, irregularities, and imprecisions of our calendars are due to things like
* the Earth's rotational period ("day")
* the Moon's orbital period ("month")
* the Earth's orbital period ("year")
not being integer multiples of each other at all! Despite that, we naturally want to keep time in terms of each of these -- as you said, they're all such extremely noticeable periodic phenomena. According to GNU units, the day is
24 hr = 86400 s
(which is a historical basis for defining the second, though not the method we officially use now because we have clocks that are more consistent than the Earth's rotation), while the lunar month is
29 days + 12 hours + 44 minutes + 2.8 seconds = 2551442.8 s
and the tropical year is
365.242198781 day = 31556926 s
If we think of a month as a fixed whole number of days, and/or think of a year as a fixed whole number of months, and/or think of a year as fixed whole number of days, it's been known for thousands of years that we're eventually going to get in trouble astronomically and require some adjustments.