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As per this Cambridge Dictionaries page,

We use in to talk about where something is in relation to a larger area around it:

A: Where’s Jane?
B: She’s in the garden.

I’ve left my keys in the car.

We use into to talk about the movement of something, usually with a verb that expresses movement (e.g. go, come). It shows where something is or was going:

A: Where’s Jane?
B: She’s gone into the house.

Helen came into the room.

Compare:

She’s gone for a walk in the garden. (She is in the garden walking.) She walked into the garden. (She entered the garden.)

With some verbs (e.g. put, fall, jump, dive) we can use either in or into with no difference in meaning:

Can you put the milk in/into the fridge? Her keys fell in/into the canal.

However, even with a verb like put, some additional context can favor the use of one preposition over the other:

• I put my hands in/into my pockets to keep them warm.
• Slowly, he put his hand into his pocket and snuck out a folding knife.

In the second example, we're placing emphasis on the movement of the hand, so into appears to be more appropriate than just in.

As per this Cambridge Dictionaries page,

We use in to talk about where something is in relation to a larger area around it:

A: Where’s Jane?
B: She’s in the garden.

I’ve left my keys in the car.

We use into to talk about the movement of something, usually with a verb that expresses movement (e.g. go, come). It shows where something is or was going:

A: Where’s Jane?
B: She’s gone into the house.

Helen came into the room.

Compare:

She’s gone for a walk in the garden. (She is in the garden walking.) She walked into the garden. (She entered the garden.)

With some verbs (e.g. put, fall, jump, dive) we can use either in or into with no difference in meaning:

Can you put the milk in/into the fridge? Her keys fell in/into the canal.

However, even with a verb like put, some additional context can favor the use of one preposition over the other:

• I put my hands in/into my pockets to keep them warm.
• Slowly, he put his hand into his pocket and snuck out a folding knife.

In the second example, we're placing emphasis on the movement of the hand, so into appears to be more appropriate than just in.

We use into to talk about the movement of something, usually with a verb that expresses movement (e.g. go, come). It shows where something is or was going: http://dictionary.cambridge.org/grammar/british-grammar/in-into

Example by @Fard:

• I put my hands in my pockets to keep them warm.
• Slowly, he put his hand into his pocket and snuck out a folding knife.

In "Put your hands into your pockets," we're talking about the movement of putting your hands.

As per this Cambridge Dictionaries page,

We use in to talk about where something is in relation to a larger area around it:

A: Where’s Jane?
B: She’s in the garden.

I’ve left my keys in the car.

We use into to talk about the movement of something, usually with a verb that expresses movement (e.g. go, come). It shows where something is or was going:

A: Where’s Jane?
B: She’s gone into the house.

Helen came into the room.

Compare:

She’s gone for a walk in the garden. (She is in the garden walking.) She walked into the garden. (She entered the garden.)

With some verbs (e.g. put, fall, jump, dive) we can use either in or into with no difference in meaning:

Can you put the milk in/into the fridge? Her keys fell in/into the canal.

However, even with a verb like put, some additional context can favor the use of one preposition over the other:

• I put my hands in/into my pockets to keep them warm.
• Slowly, he put his hand into his pocket and snuck out a folding knife.

In the second example, we're placing emphasis on the movement of the hand, so into appears to be more appropriate than just in.

We use into to talk about the movement of something, usually with a verb that expresses movement (e.g. go, come). It shows where something is or was going: http://dictionary.cambridge.org/grammar/british-grammar/in-into

Example by @Fard:
I put my hands in my pockets to keep them warm. -- Slowly, he put his hand into his pocket and snuck out a folding knife.

"Put your hands into your pockets.": We're talking about the movement of putting your hands.

We use into to talk about the movement of something, usually with a verb that expresses movement (e.g. go, come). It shows where something is or was going: http://dictionary.cambridge.org/grammar/british-grammar/in-into

Example by @Fard:

• I put my hands in my pockets to keep them warm.
• Slowly, he put his hand into his pocket and snuck out a folding knife.

In "Put your hands into your pockets," we're talking about the movement of putting your hands.