Potential advisor offering to fund visit to lab I'm planning on travelling to another country to meet a potential PhD advisor (we're both in Europe). He has said his lab will cover the expenses. 

Is this usual practice? I'm from a country where standard etiquette when offered money (or indeed, anything) in a circumstance like this is to politely refuse the offer multiple times, even when they insist, and then eventually agree while making clear you feel terrible for accepting their money. Agreeing immediately would be seen as extremely rude. 

I'm aware it's almost certainly the lab's budget that will be funding my visit and not coming out of the guy's own pocket, but still, I don't really know what to say, a standard thank you still feels bad to me, but maybe that's just from my culture.
I do not think the etiquette you are talking about applies here. You need to go there in order to ensure compatibility between you and your potential future advisor. This is neither a gift nor a present; it it a business related trip and I do not see a reason for not agreeing to this.