It’s not 100% foolproof, but it turns out voter fraud by impersonation is very rare, so it’s good enough.
When you think in terms of “make sure every vote we count was legitimate”, then “not completely foolproof” becomes a solid argument for voter ID.
Instead, if you take a wider view and think in terms of “getting the best quality estimate of the will of the voting population”, the argument against requiring ID (in the US at least) is that it would distort the results of the election far more than a tiny amount of undetected impersonation fraud does.
This will vary by country. In the US, there are barriers to getting ID for some groups (you need to go in person during business hours, pay and wait an unknown amount of time, and this needs to happen weeks ahead of election days; this is a barrier to someone without transport juggling multiple jobs and childcare, for instance.)
Other countries see the trade off differently, or use different fraud prevention approaches. For example, I know India uses indelible ink stains on fingers to prevent multiple voting, and in the UK, there is no ID requirement (yet) but the ballots are serialised and the secrecy of the ballot can be broken to investigate fraud allegations. Neither of these approaches would be culturally acceptable in the US.