Bitte steck das Papier mit einer Stecknadel fest.

Because this is an imperative sentence, meaning a command or instruction.

In German, the subject is usually left out in the informal singular imperative:

• Du steckst das Papier fest. = you pin the paper in place.
• Steck das Papier fest! = pin the paper in place!

So the du is understood and does not need to be said.

Steck is the informal singular imperative form of stecken.

For many verbs, the du-imperative is made from the du form by removing -st:

• du steckst → steck!

You may also sometimes see stecke!, but steck! is the more usual modern form here.

Compare:

• Steck das Papier fest. = informal, to one person
• Steckt das Papier fest. = informal, to more than one person
• Stecken Sie das Papier fest. = formal/polite

Because this is a separable verb: feststecken.

In German main clauses and imperatives, the separable part goes to the end:

• Ich stecke das Papier fest.
• Bitte steck das Papier fest.

But in the infinitive, it stays together:

• das Papier feststecken
• Ich möchte das Papier feststecken.

So steck ... fest is the normal pattern.

Fest adds the idea of securely, firmly, or in place.

So stecken by itself can mean something like put/stick/insert, but feststecken means more like:

• pin in place
• fasten with a pin
• secure firmly

That is why fest is important here. Without it, the sentence would sound less natural or would suggest a different action.

Papier is a neuter noun, so its basic article is das:

• das Papier

Here it is the direct object of the verb, so it is in the accusative case. For neuter nouns, the definite article is the same in nominative and accusative:

• nominative: das Papier
• accusative: das Papier

So there is no visible change.

Because mit always takes the dative case.

Stecknadel is feminine:

• nominative: eine Stecknadel
• dative: einer Stecknadel

So after mit, you must say:

• mit einer Stecknadel

This phrase tells you the instrument used to do the action: with a pin.

Because German very often forms compound nouns by joining words together.

Stecknadel is made from:

• stecken = to stick / pin
• Nadel = needle

So Stecknadel literally means something like sticking needle, but in normal English it is simply pin.

German does this all the time:

• Haus
  • Tür → Haustür
• Schreib
  • Tisch → Schreibtisch

In German compounds, the last part decides the gender.

The last part here is Nadel, and Nadel is feminine:

• die Nadel

So the whole compound is also feminine:

• die Stecknadel

This is a very useful rule for compound nouns in German.

Because all nouns are capitalized in German.

So:

• das Papier
• die Stecknadel

Also, Bitte is capitalized here only because it is the first word of the sentence. If it appeared in the middle, it would normally be lowercase:

• Bitte steck das Papier fest.
• Steck bitte das Papier fest.

Bitte is quite flexible. It often acts as a polite softener in commands.

All of these are possible:

• Bitte steck das Papier mit einer Stecknadel fest.
• Steck bitte das Papier mit einer Stecknadel fest.
• Steck das Papier bitte mit einer Stecknadel fest.

The first version sounds very natural and straightforward.

So bitte does not change the grammar much here; it mainly makes the instruction more polite.

It is informal, because of the imperative steck, which is used with du.

If you wanted to say the same thing formally or politely to one person, you would say:

• Bitte stecken Sie das Papier mit einer Stecknadel fest.

So:

• steck = informal singular
• stecken Sie = formal/polite

This is an important distinction in German commands.