In short: switching bank account is a legal service in which, with your written authorisation, the new bank moves your payments, balance and standing orders from the old account and, if you wish, closes it.
Simply put: You are not tied to one bank for ever, and you don't have to notify your employer and everyone you pay — the new bank organises the switch. It is worth it when the yearly saving beats the effort and any closing fees.
Example: Your current account costs €70 a year; a similar one at another bank costs €30. The saving is €40 a year.
Watch out: the old account is closed only if nothing is owed on it.
The EU rule is the switching service in Directive 2014/92/EU, with fixed deadlines and compensation for losses caused by the banks.
Example: Bulgaria — Payment Services and Payment Systems Act, Articles 111–115: the new bank asks the old one for the information within 2 working days of your authorisation; the old bank sends it within 5 working days; the new bank sets up the payments within 5 working days. Information on standing orders is free (Article 113(1)), and losses from a breach are refunded without delay (Article 114).
Check in your country — your bank's switching information leaflet.
Source: Directive 2014/92/EU: http://data.europa.eu/eli/dir/2014/92/oj · Payment Services and Payment Systems Act, Articles 111–115 — consolidated text (in Bulgarian): https://kik-info.com/normativna-baza/zakoni/zpups/