В холодильнике остались сосиски, кетчуп и кусок сыра.

Because в холодильнике expresses location: the items are in the fridge.

Russian uses:

• в + accusative for motion into something
  
  • положить в холодильник = to put into the fridge
• в + prepositional for being inside something
  
  • в холодильнике = in the fridge

So here, nothing is moving. The sentence describes where the items are, so в холодильнике is correct.

It is in the prepositional case.

The dictionary form is холодильник. After в meaning in (location), it becomes:

• холодильник → в холодильнике

This is a very common pattern:

• в доме = in the house
• в магазине = in the store
• в холодильнике = in the fridge

Остались comes from остаться, which means to remain, to be left, or to stay behind.

In this sentence, it means something like:

• there were still ... left
• ... remained in the fridge

So the sentence is not just saying the items are in the fridge. It suggests that these are the things that are still left there.

That is why остались is more natural here than just были in many contexts.

Because it agrees with the subject, and the subject is a list of several things:

• сосиски
• кетчуп
• кусок сыра

Together, that is a compound subject, so Russian normally uses a plural verb.

That is why you get:

• остались = remained / were left (plural)

not

• остался = singular masculine
• осталась = singular feminine
• осталось = singular neuter

Russian word order is flexible, and this order is very natural when introducing or presenting what exists somewhere.

В холодильнике остались сосиски, кетчуп и кусок сыра feels like:

• In the fridge, there were still sausages, ketchup, and a piece of cheese left.

This structure often puts:

1. the place / setting first
2. the verb
3. the things being mentioned

It sounds natural because the sentence is answering something like:

• What was left in the fridge?

You could rearrange the words, but the emphasis would shift.

Because they refer to different kinds of nouns:

• сосиски = sausages — here, more than one sausage
• кетчуп = ketchup — an uncountable/mass noun, treated as singular
• кусок сыра = a piece of cheese — one piece

So the sentence mixes:

• a plural count noun
• a singular mass noun
• a singular count expression

That is completely normal.

Because after кусок (a piece / a chunk), the thing being measured usually goes in the genitive case.

So:

• кусок сыра = a piece of cheese
• кусок хлеба = a piece of bread
• стакан воды = a glass of water
• бутылка молока = a bottle of milk

Here:

• dictionary form: сыр
• genitive singular: сыра

So кусок сыра is the normal Russian pattern for a piece of cheese.

They are all functioning as the subject of the sentence, so they are in the nominative case.

More precisely:

• сосиски = nominative plural
• кетчуп = nominative singular
• кусок = nominative singular
• сыра is genitive, but only because it depends on кусок

So the whole phrase кусок сыра is part of the subject, even though inside that phrase сыра is genitive.

Yes. Остались is past tense, plural.

You can break it down like this:

• оста- / оста- = verb stem
• -л- = past tense marker
• -и = plural ending
• -сь = reflexive ending

So остались literally means remained / were left.

Russian past tense agrees in gender and number, not person:

• остался = masculine singular
• осталась = feminine singular
• осталось = neuter singular
• остались = plural

It is the reflexive marker of the verb остаться.

Many Russian verbs exist in both non-reflexive and reflexive forms, and the reflexive form often has a different meaning.

For example:

• оставить = to leave something
• оставаться / остаться = to remain / to be left

So:

• Я оставил кетчуп в холодильнике = I left the ketchup in the fridge
• В холодильнике остался кетчуп = There was ketchup left in the fridge

In this sentence, the reflexive form is the one you need because the meaning is to remain / be left.

Because Russian has no articles.

English needs to choose between:

• a piece of cheese
• the ketchup
• some sausages

Russian usually leaves that information to context.

So:

• сосиски could mean sausages, some sausages, or the sausages
• кетчуп could mean ketchup, some ketchup, or the ketchup
• кусок сыра could mean a piece of cheese or sometimes the piece of cheese, depending on context

The exact English article depends on the situation, not on a separate Russian word.

Sometimes yes, but the meaning would be a little different.

• В холодильнике есть сосиски, кетчуп и кусок сыра = There are sausages, ketchup, and a piece of cheese in the fridge
• В холодильнике остались сосиски, кетчуп и кусок сыра = There are sausages, ketchup, and a piece of cheese left in the fridge

So есть simply states existence, while остались adds the idea that these items remain or are still left.

That extra nuance is important. The original sentence suggests that maybe other food is gone, but these things are still there.