Reverse charge moves the VAT-paying responsibility from you to your customer in another EU country. The responsibility can only be moved from business to business.
You’ll write an invoice/receipt without VAT (or 0% VAT) and include a text “Reverse charge, VAT directive art. 44” and you are done. In practice the text is often missing, as people re-use the same invoice format they use for non-EU sales.
My British corporation pays an American company for Web hosting. However, the US company does not charge us VAT, so we have to charge ourselves ("reverse charge") the VAT at UK rates and then immediately claim back the VAT as it is used in our VATable supplies. The net result is zero but reverse charging means they get VAT revenue for situations where VAT is not reclaimable (many exist).
Another example is that my British corporation sells services to a, say, French corporation. We charge no VAT on this sale but the French corporation has to then "reverse charge" French VAT on the sale within France (and then claim it back, if valid). We then also have to file this transaction on an "EC Sales List" so that the various authorities can check that the reverse charge did occur.