На решётке кабачок поджарится быстрее, чем баклажан.

Because на can mean either on/onto depending on whether you are talking about location or movement.

• на решётке = on the grate / on the grill rack → location, so Russian uses the prepositional case
• на решётку = onto the grate → movement, so Russian uses the accusative case

Here the food is already there, so на решётке is the correct form.

Here решётка means a grate, rack, or grill grid—the metal lattice you cook on.

So на решётке is not just a literal on a grid in some abstract sense; in cooking context it means on the grill grate / on the rack.

Russian often uses the singular to talk about a food item or a whole category in a general way, much like English can say zucchini cooks faster than eggplant.

So кабачок here is the subject in the nominative singular. It does not have to mean one specific zucchini only; it can be a general statement about zucchini as an ingredient.

Historically, yes, it has a diminutive-looking form, but in modern Russian кабачок is the normal neutral word for zucchini / summer squash.

So in this sentence it does not sound especially cute or childish. It is just the ordinary vegetable name.

Поджарится is perfective future.

That is a very important Russian pattern:

• imperfective future = будет + infinitive
• perfective future = a simple one-word future form

Here the verb is поджариться (perfective), so the future is simply поджарится.

The perfective aspect focuses on the result: the zucchini will get browned / cook through to that point.

Not literally. In sentences like this, -ся often makes the verb more intransitive or middle/passive-like.

So:

• поджарить = to fry/brown something
• поджариться = to get fried / to brown / to cook until browned

The idea is not that the zucchini actively does it to itself, but that it undergoes the change.

Because after чем in this kind of comparison, Russian often treats the second part like an omitted clause:

• ... быстрее, чем баклажан [поджарится]

So баклажан stays in the nominative, because it is understood as the subject of the omitted verb.

There is also another comparison pattern without чем:

• кабачок поджарится быстрее баклажана

That version uses the genitive (баклажана). Both patterns are possible, but this sentence uses the чем + nominative type.

Because Russian very often leaves out repeated words when they are easy to understand from context.

The full version would be something like:

• На решётке кабачок поджарится быстрее, чем баклажан поджарится.

But that sounds unnecessarily repetitive. Russian prefers the shorter version.

Because чем introduces the second part of the comparison, and in this structure Russian normally separates it with a comma:

• быстрее, чем баклажан

So the comma is standard punctuation here.

Yes. Russian word order is flexible.

This sentence starts with На решётке to set the scene first: on the grate / on the grill.

Other orders are possible, for example:

• Кабачок на решётке поджарится быстрее, чем баклажан.

That would still be grammatical, but the emphasis shifts a little. The original version highlights the cooking location right away.

The main stress points are:

• на решётке → re-SHYOT-ke
• кабачок → kaba-CHOK
• поджарится → pod-ZHAR-it-sya
• быстрее → by-STRE-ye
• баклажан → bakla-ZHAN

A useful tip: ё is always stressed, so in решётке the stress must fall there.