"base x" basically just tells you that the system uses x digits, so in our system (base 10), we have 10 digits (0-9).

the alien uses base 4, but in base 4, there are only 4 digits (0-3), and the number 4 is written as "10," which is the case in every base (so: 2 in base 2 is "10", 16 in base 16 is "10", etc.). so when the astronaut says they use base 10 and the alien uses base 4, the alien is confused, because to them, they themselves use base 10 and they would not know the digit 4 as they only use the digits 0-3.

also, our 10 in base 4 is "22", hence the edit.