Another, perhaps more anecdotal than maths, perspective.
If you've come across the metaphor "induction is like dominoes", showing "P(k+1) is true assuming P(k) is true" is like you've built the perfectly aligned domino line. But no one will be convinced your domino line actually works until everything gets toppled. Showing a base case is true is to actually initiate the toppling sequence somewhere - usually from the beginning (i.e. P(1) is true).
Now perhaps less anecdotal, but still not quite mathy example (mostly it's not really well-posed, but meh).
Suppose you want to show that all horses in the world have the same color. Well, the induction step starts by assuming if you have k horses, they will have the same color. Then in the k+1 case, the first k horses have the same color by assumption, and the last k horses also have the same color by assumption, so all k+1 horses do have the same color... The induction step is perfectly fine, but our base case is not true (e.g. n=2 obviously fails).
Sidenote: Obviously all the base case check you do in A-Levels are painfully trivial. But apparently there are loads of problems out there where the induction step is actually the easy stuff, and the base case check is the actual hard stuff. God knows what the problems are - I'm not that good at maths yet.