Na pěšině je bláto.

Because pěšině is the locative singular form of pěšina.

The dictionary form is pěšina = path / trail.
After the preposition na when it means on / at in the sense of location, Czech uses the locative case.

So:

• pěšina = base form
• na pěšině = on the path / on the trail

This is a very common pattern in Czech.

Because na can take different cases depending on the meaning.

• na + locative = location, where something is
• na + accusative = direction, where something is going

Here the sentence describes a location, not movement:

• Na pěšině je bláto. = mud is on the path
• Jdu na pěšinu. = I am going onto/to the path

So na pěšině is correct because the sentence answers Where is the mud?

Czech often uses na with surfaces, open areas, routes, and some kinds of places where English might use on, in, or even at depending on context.

A pěšina is a path/trail, so Czech naturally says:

• na pěšině

Using v here would sound unnatural in standard Czech.
For a learner, the safest approach is to learn this as a fixed combination:

• na pěšině = on the path

Je is the 3rd person singular present form of být = to be.

So:

• je = is

In this sentence it links the location and the thing being talked about:

• Na pěšině je bláto. = There is mud on the path. / Mud is on the path.

Unlike in some languages, you normally do need this verb in standard Czech in the present tense.

Bláto is in the nominative singular because it is the grammatical subject of the sentence.

The basic form is:

• bláto = mud

It is a neuter noun, which is why the verb is singular:

• bláto je = mud is

So nothing special is happening to bláto here; it is just in its basic dictionary form.

Yes, you could also say Bláto je na pěšině. Both are grammatical, but the emphasis is different.

• Na pěšině je bláto.
  This sounds more like introducing what is on the path. It is a very natural way to present new information.

• Bláto je na pěšině.
  This puts more focus on bláto as the topic.

Czech word order is more flexible than English, and it often changes according to information structure: what is already known and what is new or important.

In natural English, it is often best translated as:

• There is mud on the path.

Czech commonly uses sentences with je in a way that corresponds to English there is/there are, even though Czech does not need a separate word like there in this structure.

So while the literal structure is closer to On the path is mud, the natural meaning is usually:

• There is mud on the path.

Because Czech has no articles like English a/an/the.

So:

• pěšina can mean a path or the path
• bláto can mean mud, some mud, or the mud, depending on context

The exact meaning is understood from the situation, context, and word order.

A rough pronunciation guide is:

• pě sounds approximately like pye
• š sounds like sh
• ni here is softened because of ě
• final -ě is not pronounced like a separate strong syllable in the English sense; it helps indicate palatalization/softening

A rough English-friendly approximation is:

• PYE-shi-nye

More carefully:

• pě = softened p
• ši = shi
• ně = softened nye

Also remember that Czech stress is almost always on the first syllable:

• PĚ-ši-ně

The letter ě often signals that the preceding consonant is pronounced in a softened way.

In this word:

• pě is not just pe
• ně is not just ne

For English speakers, it often helps to think of ě as creating a slight y-like quality after the consonant:

• pě ≈ pye
• ně ≈ nye

This is not a perfect rule for every Czech word, but it is a useful beginner guideline.

Usually bláto is treated as an uncountable mass noun, like mud in English.

So Czech normally says:

• je bláto = there is mud

You would not usually use a plural unless talking about different kinds or occurrences in a special context. In everyday speech, singular is the normal choice.

In standard Czech, not in this sentence.

You need:

• Na pěšině je bláto.

Leaving out je would sound incomplete or nonstandard here.

Czech does omit to be in some special styles or headings, but in an ordinary full sentence like this, je should stay.