Cut Into / Cut into line

We use cut into when we want to cut into something, but we can also use it in a metaphoric way. 

For example, if we say 'I want to cut into this nice piece of meat', it's more direct and apparent than if we said 'This lady always cuts into us'

The second example means to interrupt.

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Let's look at a few examples

People always like to cut into line so they can be the first. They don't care how long you were staying there.....

What if we're driving and someone wants to get into our lane. We can say

I'll slow down to let this lady cut in. 

We can see from this video how cutting into someone can cause an accident. Some people should stick to driving animals. It's much safer and eco friendly.

 Also when you're speaking, and someone interrupts you, you can say that person cuts in when I am talking.