Let's say that you have business where you put together and sell computers. If you worked out of Dallas and sold a computer to someone in Houston, you'd collect the tax and send it to the state. The state then keeps a portion and sends a portion to the local governments where you live.
Now, let's say that you move to some other state and sell a computer to someone in Houston. Normally, if you mail it directly to your customer, you do not collect any tax and it is up to your customer to pay a local use tax.
Amazon's problem is that it has a distribution centre in the Dallas area, and the state says that counts as a 'local presence', but Amazon does not agree.
To carry on with the example, you live out of state and you sell a computer to someone in Houston. However, instead of shipping it directly to the customer, your brother who visits every weekend picks up the computer, gets it to his home in Dallas, and then ships it out from there. The state finds out about this and decides that your brother counts as a "local presence". Accordingly, it then demands that you pay up on all of the taxes you have failed to collect.
They want to open a facility in Texas, but they don't want to have to start charging sales tax. Doesn't hurt to ask.
What the public in general seems to miss is that the tax revenue is due one way or another - either Amazon collects or customers self pay. Most of the public who are not business owners do not realize they need to self pay - it is neither enforced nor made obvious that one needs to do so. So instead of making the public aware of 'use tax' and trying to enforce it, the Texas legislature is trying to change the definition of an in-state presence to include independent subsidiaries, affiliates, etc. Then by doing so, the government can force the merchant to collect the taxes owed for them.
The first 'internet sales tax' bill was passed by the legislature but vetoed by the governor. Similar language was added to an omnibus spending bill, which is much more likely to survive.