Las pilas de repuesto son para la radio a pilas.

Because pila and radio are both feminine nouns in Spanish.

• la pila → singular
• las pilas → plural
• la radio → singular

So the articles have to agree in gender and number with the nouns they go with.

Because the subject of the sentence is Las pilas de repuesto, which is plural.

• Las pilas de repuesto = the spare batteries
• son = are

Even though la radio is singular, it is not the subject. It comes after para and just tells you what the batteries are for.

De repuesto means spare or replacement.

So:

• pilas de repuesto = spare batteries / replacement batteries

This is a very common fixed expression in Spanish:

• una rueda de repuesto = a spare wheel/tyre
• piezas de repuesto = spare parts

Because de repuesto is not a simple adjective like grande or rojo. It is a phrase with de, and in Spanish these kinds of descriptive phrases normally come after the noun.

So Spanish says:

• pilas de repuesto

not normally:

• de repuesto pilas

This is just the natural word order.

A pilas means battery-powered or running on batteries.

In this sentence:

• la radio a pilas = the battery-powered radio / the radio that uses batteries

The preposition a is often used to show how something works or what it runs on:

• a gas = gas-powered / on gas
• a mano = by hand
• a motor = motor-powered

So a pilas is a set expression.

For the meaning radio as a device or medium, Spanish normally uses la radio.

• la radio = the radio
• el radio usually means radius in maths/geometry

So in this sentence, la radio is the correct form.

In Spain, pilas usually refers to the small batteries you put into devices like radios, remote controls, or toys.

Batería often refers to:

• a larger battery
• a rechargeable battery pack
• a car battery
• or even a drum kit

So for a portable radio, pilas is the most natural choice in Spain.

The two instances of pilas do different jobs:

• Las pilas de repuesto = the spare batteries themselves
• la radio a pilas = the battery-powered radio

So even though the same word appears twice, it is not redundant in Spanish. It sounds natural.

Because the usual expression is a pilas, in the plural, when talking about devices powered by batteries.

Spanish treats this as a standard phrase:

• una linterna a pilas
• un reloj a pilas
• una radio a pilas

Even if a device only uses one battery, Spanish still often uses a pilas as the normal expression.

Yes, you could, but the meaning changes slightly.

• para la radio a pilas suggests a specific radio, probably one already known in the conversation
• para una radio a pilas means for a battery-powered radio in a more general sense, or for some unspecified radio

So la is more specific, while una is less specific.