I have tried my best to find an info about this on the internet, but had no luck, sadly. So I would ask here:

Please take a look on this [spam link removed]

Can anybody tell me please how to make the same contact form (at the bottom-right corner)?

I assume you mean the realtime chat/messaging functionality which iis a free script to implement from https://www.tawk.to/

But the link (and anchor text you used) to that site smells spammy to me too.