I went through exactly the same thing a while back and it's not as complicated as it seems. The electrician is right about the 32 amps, but before you freak out about major work, ask them to check your electrical panel and see what power capacity you have contracted right now - a lot of times you can get away with just upgrading your contract with the utility company without touching practically anything in your house, just replacing a breaker or the main switch. If your panel is old or already maxed out with other appliances, then yeah, you'll need to do some work, but the electrician will see that clearly after reviewing everything properly.
The important thing is not to install the wallbox without sorting that out first, because otherwise you'll be right on the edge of tripping the breaker and it's dangerous besides. I got quotes from a couple of installers and the costs varied a lot depending on whether I needed to run new wiring or not - without major work it was in a reasonable price range, but if you needed to upgrade the service entrance it would double. I asked for everything in writing with the panel inspection included, so I knew exactly what each part would cost me before I signed off on anything.