I'm a new player who decided to play as a Bard, but I noticed that once you reach level 3, you get to choose a college.
How should I explain my character suddenly getting a college education in the middle of the game? Are there any special preparations I should do in my character sheet, like adjusting my background for it?
A "college education" means something specific in the modern world: it means that someone has gone to a specific kind of educational institution to get a specific kind of education.
A bardic college is not a modern college. Instead, it's basically a club (PHB 54):
Bards form loose associations, which they call colleges, to facilitate their gatherings and preserve their traditions.
Consider the most learned-sounding college, the College of Lore:
The college’s members gather in libraries and sometimes in actual colleges, complete with classrooms and dormitories, to share their lore with one another. They also meet at festivals or affairs of state, where they can expose corruption, unravel lies, and poke fun at self- important figures of authority.
Notice how the first sentence draws a distinction between the bardic college and "actual colleges"?
Your character's progression through levels is how you explain "getting a college education". As they spend time hanging out with people in their college, they learn more and more and gain better and better abilities, which is represented game-wise by your class levels.
To supplement Icyfire's existing (very good) answer:
While that is not written down in the rules as such, I have always assumed that new abilities gained on a level-up don't just appear from nothing one morning, but are the result of continuous training, study and exercise in the time leading up to that level. The level-up is finally the moment where your character is proficient enough to reliably use these abilities, even in stressful situations (e.g. combat).
In the case of your bard, this could mean that your membership in the “college“ of your choice is officially recognized once you reach level 3, although you have been studying and practicing their techniques for a long time.
... then make the case that the bard has already been to college, but is having to finish her final degree granting project during a co-op or work study program. You can similarly compare it to having submitted their thesis project and having been employed to pay the rent (adventuring) before The Board finally grants the degree to the bard after confirming that all requirements were met1.
Apply that to your bard in-game in a way that fits the campaign. Your bard has been studying and doing labs for some time, and is now out 'in the field' applying what they learned in the classroom. The results of their combined efforts, class room and practical, combine to fulfill degree requirements so that they are declared members in good standing of this college. (An old bard alumnus/sponsor has been scrying on the PC bard from time to time, since magic exists, and sees the work in progress).
There are a myriad of ways to make this rich and colorful through RP ... such as ...
A message arriving via courier: your degree has been granted!
A message arriving via the sending spell form a higher level bard or a cleric: in 25 words or less, you got the degree and here's how to do the secret handshake
The player and DM don't have to confine the level up at 3rd level to "Hey look, enough points for 3rd level, ding!" if they don't want to. (But that is certainly an option)
Have fun with this. Make it a big deal, a cause for celebration.
1 As a real life example: I completed a course of study in late 1988 for a master's degree, but I moved (had to thanks to my job) before the final package / project was forwarded to the University (we were a satellite campus) for approval and acceptance. In May of 1989 the final confirmation paperwork came in; degree granted.
