“Listen, I don't want to take up too much of your time.”
You're making a sales call to a potential customer. After introducing yourself, you want to make the customer feel relaxed. You let her know that your sales pitch will be short.
Listen, I don't want to take up too much of your time.
To "take up" something means to use it. This phrasal verb is most often used with "space" and "time":
It takes up too much space.
Having kids takes up almost all of your free time.
This would not work with something like a tool. "Can I take up your hammer?" would be incorrect, for example.
When you start off a conversation with "Listen", it seems that you want to get to the main point of the conversation quickly. You use it like this:
Listen, I know that we've had disagreements in the past, but I hope that we can work together on this.
Listen, we don't have much time, so let me explain what happened.
You can use the phrase "I don't want to take up too much of your time" at the beginning of a conversation, to show that the conversation is going to be short:
Hey, Melanie, I don't want to take up too much of your time, but I need to go over these sales projections with you. Do you have a second?
You can also use this phrase to end a conversation politely, in the same way as the similar phrase "I don't want to take up any more of your time":
Listen, I don't want to take up too much of your time, but it was great talking to you.