Kan du stänga fönstret, annars blir sovrummet kallt.

Literally, Kan du … means Can you …, but in everyday Swedish it’s very commonly used as a polite request, much like English Could you … / Would you … depending on tone.
If you want to sound a bit more formal/softer, Swedish often uses:

• Kan du vara snäll och stänga fönstret? (Can you please close the window?)
• Skulle du kunna stänga fönstret? (Could you close the window?)

Because after a modal verb like kan, Swedish uses the infinitive form of the main verb:

• kan + infinitive → kan stänga (can close)

stäng is the imperative (command form):

• Stäng fönstret! = Close the window!

So your sentence is a request with a modal, not an imperative.

fönster = window (an ett-word, i.e., neuter gender)
fönstret = the window (definite form)

Swedish often attaches the definite article to the end of the noun:

• ett fönster = a window
• fönstret = the window
• fönster = windows
• fönstren = the windows

annars most often means otherwise / or else. It introduces the consequence if the first action doesn’t happen.

In your sentence it’s placed after a comma and works like a linking adverb:

• Kan du stänga fönstret, annars blir sovrummet kallt.
  = Close it; otherwise the bedroom will get cold.

You can also start a new sentence with it:

• Kan du stänga fönstret? Annars blir sovrummet kallt.

Both are possible, but annars triggers the common Swedish V2 word order rule: the finite verb typically comes in the second position in a main clause.

After annars, Swedish often puts the verb first and the subject after it:

• Annars blir sovrummet kallt. (Otherwise the bedroom gets cold.)

Without annars (or with a simpler start), you get the more “neutral” subject-first order:

• Sovrummet blir kallt. (The bedroom gets cold.)

• är = is (a state/condition)
• blir = becomes / gets (a change into a state)

So:

• Sovrummet är kallt. = The bedroom is cold (already cold).
• Sovrummet blir kallt. = The bedroom gets cold (it turns cold, as a result).

In your sentence, blir fits because leaving the window open causes a change.

sovrum is also an ett-word:

• ett sovrum = a bedroom
• sovrummet = the bedroom
• sovrum = bedrooms
• sovrummen = the bedrooms

Here, sovrummet refers to a specific bedroom (contextually “the bedroom”).

Because Swedish adjectives agree with the gender/number of the noun they describe.

sovrummet is an ett-word (neuter), so the predicate adjective takes -t:

• Ett sovrum är kallt. (A bedroom is cold.)
• Sovrummet blir kallt. (The bedroom gets cold.)

For an en-word you’d use the basic form:

• En lägenhet är kall. (An apartment is cold.)

Plural uses -a:

• Rummen är kalla. (The rooms are cold.)

You can write it either way. Swedish often uses a comma here to connect two closely linked main clauses, but it’s also very natural to split them:

• With comma: Kan du stänga fönstret, annars blir sovrummet kallt.
• Two sentences: Kan du stänga fönstret? Annars blir sovrummet kallt.

Both are common; two sentences can feel slightly clearer in writing.

A rough guide (dialects vary):

• stänga: the ä is like a front vowel similar to the vowel in English bed, but usually a bit “brighter/tenser” in Swedish. The ng is like in English sing.
• fönstret: ö is a rounded front vowel (not found in most English accents). Many learners approximate it by shaping the lips for o while trying to say e.
• sovrummet: o here is often a long vowel (depending on accent) and the u in rum is also a Swedish vowel that doesn’t match English u well (it’s more fronted).

If you want, tell me which accent you’re aiming for (Stockholm/Standard Swedish vs Finland Swedish, etc.) and I can give more precise IPA.