Я вытираю стол влажной тряпкой.

Russian has no articles, so there is no separate word for the or a/an.

So стол can mean the table or a table depending on context. Russian relies on context, word order, and the situation to show whether something is specific or not.

Вытираю is the 1st person singular present tense form of the verb вытирать.

That means:

• я вытираю = I wipe / I am wiping

The ending -ю tells you the subject is I, which is one reason Russian can often omit я.

It can mean either one.

Russian present tense does not make the same obligatory distinction that English makes between:

• I wipe
• I am wiping

So Я вытираю стол влажной тряпкой can mean:

• I wipe the table with a damp cloth
• I am wiping the table with a damp cloth

If you want to make it clearly right now, you can add something like сейчас:

• Сейчас я вытираю стол влажной тряпкой.

Because стол is the direct object, so it is in the accusative case.

However, for masculine singular inanimate nouns, the accusative looks exactly the same as the nominative.

So:

• nominative: стол
• accusative: стол

That is why the form does not change.

Compare that with an animate noun:

• брат → я вижу брата

There, the accusative is different.

Because they are in the instrumental case.

Russian often uses the instrumental case to show the tool or means used to do something.

Here, the cloth is the tool:

• тряпка = cloth / rag
• тряпкой = with a cloth

The adjective must agree with the noun, so:

• влажная тряпка = a damp cloth
• влажной тряпкой = with a damp cloth

So both words change because they are feminine singular instrumental.

Because with an instrument, Russian usually uses the instrumental case by itself, with no preposition.

So:

• Я вытираю стол влажной тряпкой = I wipe the table with a damp cloth

This is the normal Russian structure.

Using с here would usually be wrong or would change the meaning. С often means together with or accompanied by, not simply using as a tool.

Yes, very often.

Because вытираю already means I am wiping / I wipe, Russian does not always need the pronoun.

So both are possible:

• Я вытираю стол влажной тряпкой
• Вытираю стол влажной тряпкой

Including я can add emphasis, contrast, or clarity.

Yes. Russian word order is fairly flexible because the case endings show the grammatical roles.

For example, these are all possible:

• Я вытираю стол влажной тряпкой.
• Стол я вытираю влажной тряпкой.
• Влажной тряпкой я вытираю стол.

The basic meaning stays the same, but the focus changes:

• starting with стол emphasizes the table
• starting with влажной тряпкой emphasizes the cloth

The version you were given is the most neutral/basic order.

Влажный usually means damp or moist.
Мокрый means wet.

So:

• влажная тряпка = a damp cloth
• мокрая тряпка = a wet cloth

In this sentence, влажная sounds very natural, because when you wipe a table, you often use a cloth that is damp rather than soaking wet.

Вытирать is imperfective.

That means it focuses on:

• the process
• repeated action
• an action without emphasizing completion

Its perfective partner is вытереть.

So:

• Я вытираю стол влажной тряпкой = I am wiping / I wipe the table with a damp cloth
• Я вытру стол влажной тряпкой = I will wipe the table with a damp cloth
• Я вытер стол влажной тряпкой = I wiped the table with a damp cloth

A very important point: Russian normally does not use a true present tense form of a perfective verb for present-time meaning. Perfective present forms usually have future meaning.

The usual stress is:

Я вытира́ю стол вла́жной тря́пкой.

The stressed syllables are:

• вытира́ю
• вла́жной
• тря́пкой

A rough pronunciation guide:

• вытира́ю ≈ vee-tee-RA-yu
• вла́жной ≈ VLAZH-noy
• тря́пкой ≈ TRYAP-koy

Also remember that unstressed Russian vowels are often reduced in pronunciation, so the spoken form may sound less like the spelling than English learners expect.