Proszę jechać ostrożnie, bo ulica jest śliska.

Proszę + infinitive is a very common polite way to give an instruction or request in Polish.

• Proszę jechać ostrożnie = Please drive/go carefully
• Jedź ostrożnie = Drive carefully

The version with proszę sounds more polite, less direct, and is often used:

• in formal situations
• by service workers
• in announcements
• when speaking respectfully to someone

So Proszę jechać ostrożnie is more like a polite instruction, while Jedź ostrożnie is a plain imperative.

Jechać usually means to go/travel by some vehicle:

• by car
• by bus
• by train
• by bike
• sometimes more generally when movement involves transport

So in this sentence it suggests driving/riding/travelling carefully, not walking.

This differs from:

• iść = to go on foot
• chodzić = to walk / go habitually
• jeździć = to travel/ride repeatedly or habitually

Here jechać is the right choice because the warning fits movement in a vehicle on a slippery street.

Because ostrożnie is an adverb, and the sentence needs an adverb to describe how someone should travel.

• ostrożny = careful (adjective, used to describe a person or noun)
• ostrożnie = carefully (adverb, used to describe an action)

Compare:

• On jest ostrożny. = He is careful.
• On jedzie ostrożnie. = He is driving carefully.

In this sentence, we are describing the action jechać, so the adverb ostrożnie is required.

Bo means because.

So:

• Proszę jechać ostrożnie, bo ulica jest śliska. = Please drive carefully, because the street is slippery.

Compared with other words:

• bo = very common, natural, conversational
• ponieważ = more formal, more written
• dlatego że = also because, often a bit heavier stylistically

All are possible here, but bo sounds the most natural in everyday speech.

Examples:

• Zostań w domu, bo pada. = Stay home, because it’s raining.
• Zostań w domu, ponieważ pada. = same meaning, more formal

Because ulica is the subject of the clause, so it appears in the nominative case.

• ulica = nominative singular = street
• jest śliska = is slippery

So:

• ulica jest śliska = the street is slippery

Other forms would be used in other grammatical roles:

• widzę ulicę = I see the street → accusative
• na ulicy = on the street → locative

Here, since the street is the thing being described, nominative ulica is correct.

Because śliska must agree with ulica.

Ulica is:

• feminine
• singular
• nominative

So the adjective must also be:

• feminine
• singular
• nominative

That gives:

• śliska

Compare:

• chodnik jest śliski = the sidewalk is slippery
  (chodnik is masculine)
• ulica jest śliska = the street is slippery
  (ulica is feminine)
• drogi są śliskie = the roads are slippery
  (plural)

Not necessarily only a car, but very often that is the natural interpretation.

Jechać covers traveling by vehicle in general, so depending on context it could mean:

• drive carefully
• ride carefully
• travel carefully

If the person is clearly in a car, English would most naturally translate it as Please drive carefully.

If the context is broader, Please travel carefully or Please go carefully may fit better.

Yes, you can think of the addressee as being understood.

Polish often leaves out subject pronouns and titles when they are obvious from context. So:

• Proszę jechać ostrożnie can imply
  Proszę pana / Proszę pani, jechać ostrożnie in meaning and politeness, though that exact structure is not usually how you would expand it naturally.

A clearer comparison is:

• Niech pan jedzie ostrożnie.
• Niech pani jedzie ostrożnie.

These explicitly refer to sir / madam and are also polite.
But proszę + infinitive is a very standard polite instruction without naming the person directly.

Yes, but the meaning changes.

• Proszę jechać ostrożnie = about this trip / this situation now
• Proszę jeździć ostrożnie = more like Please drive carefully in general / as a habit

This is the difference between:

• jechać = one ongoing trip, one instance
• jeździć = repeated, habitual, general movement by vehicle

So for a warning about a currently slippery street, jechać is the better choice.

A rough pronunciation guide:

• Proszę ≈ PRO-sheh
• jechać ≈ YEH-hach
  (the ch is like the ch in Scottish loch)
• ostrożnie ≈ o-STROZH-nyeh
• bo ≈ boh
• ulica ≈ oo-LEE-tsa
• jest ≈ yest
• śliska ≈ SHLEES-ka

Very roughly as a whole:

PRO-sheh YEH-hach o-STROZH-nyeh, boh oo-LEE-tsa yest SHLEES-ka.

A few useful sound notes:

• sz sounds like sh
• ż sounds like the s in measure
• ś is a softer sh-like sound
• j in Polish sounds like English y

Yes, Polish word order is fairly flexible, though some versions sound more natural than others.

The original:

• Proszę jechać ostrożnie, bo ulica jest śliska.

You could also hear:

• Bo ulica jest śliska, proszę jechać ostrożnie.
• Proszę ostrożnie jechać, bo ulica jest śliska.

However, the original version is the most neutral and natural.

In Polish, changing word order often affects:

• emphasis
• rhythm
• style

But the basic meaning here stays the same.