If you're trying to explain by applying a modern life analogy ...
... 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
- Reading a public notice in the local town square. (Wizard looks at public notices and yells over to the bard: Hey, look, you've matriculated!)
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.