Playing music in a commercial space is a public performance, and it needs permission from the rights holders. The part operators get wrong is that there are two distinct rights in any recorded song, and permission for one is not permission for the other.
Two rights, not one
A recording carries the rights in the sound recording itself — the specific recorded performance — and separately the rights in the underlying musical and literary work, meaning the composition and the lyrics. These are owned and administered by different parties. An establishment playing commercial recorded music generally needs to account for both.
Why this is confusing right now
The administration of these rights in India has changed in recent years, including the copyright-society registration position of the bodies involved. Because the position has shifted and is contested in places, this lesson deliberately does not state a fee or a single authority. Confirm the current requirement directly with the relevant rights organisations before you pay anyone, and get what you are buying in writing, specifying which rights it covers and for what period.
The simpler route
For most independent restaurants the pragmatic answer is a commercial background-music service whose licence explicitly includes public-performance rights for business premises. You pay one subscription, the provider handles the underlying rights, and you hold a document that says so. It is usually cheaper than licensing commercial repertoire directly and far less administrative work.
The alternative is royalty-free or originally licensed music, where the licence terms expressly permit commercial public performance. Read the licence — a great deal of "royalty-free" music is licensed for personal use only.
What not to do
Do not assume a personal streaming subscription covers commercial use. It does not; the consumer terms of the major services exclude public performance in business premises. This is the single most common compliance gap in Indian restaurants, and it is the easiest one for an enforcement visit to establish.