I can understand why the connection between these two senses of "dispose" seems so mysterious. The key to understanding it is: "to get rid of" is actually a secondary sense, even though today it is very common. The primary sense is "to put things into their proper places". The primary sense is now rare, but it still colors and explains the secondary senses.
Putting things into their proper place
When you dispose of garbage, you put it into its proper place: a garbage can, the city dump, or someplace like that. When you dispose of radioactive waste, you put it someplace very far away from people. However, throwing away garbage is only one of many ways that things can be "disposed of".
I marveled at the ability of a South China crime boss to dispose of these resources and organizational refinements with such precision. [Source]
Here, "dispose of" doesn't mean "get rid of" those resources, it means to put them to work in a coordinated manner—that is, to bring them to the proper places for performing some large, coordinated action.
In the Order Disposal Problem:
A manufacturer … may receive and dispose many production orders during a time period [but] he can dispose at most one order at any time due to resource constraints.
the word "dispose" means "to complete a job"—another way in which something can be put into its settled and proper place.
Being ready to act
Here is an example of saying that someone is disposed to act in a certain way:
Virtually all soldiers at all times have been strongly disposed to obey orders to go to war irrespective of whether the war has been just or unjust. [Source]
This calls upon another secondary sense of dispose: to put into a state of readiness to do some action—the action that one is disposed to or toward. The elements of one's mind, one's habits, one's abilities, one's resources, one's knowledge, one's surroundings, etc. are in a configuration that makes one especially ready to perform the action. This sense also occurs in sentences like this:
His generation and his origins dispose him to be a democrat. [Source]
Figuring out more senses of dispose
The word "dispose" gets used for many, widely varying meanings—more than I can list here, and indeed more than you'll find in dictionaries. Once you understand the primary meaning, it becomes a lot easier to understand how people are extending that meaning to fit different situations.
I think it helps to know the etymology, both for understanding the many senses of "dispose" and for many other words that contain the same two roots: dis- and pose. "Dis-" (in "dispose") is a Latin root meaning to break apart and spread out, as in disperse, disseminate, dissolve. "Pose" is a Latin root meaning to put something in a certain place or state, as in position, opposite, repose.
Knowing that, you might be able to figure out that a disposition can mean both an inclination toward doing something and the result of doing something—"where" things ultimately got put. The latter sense is rare, but an example is that in computers, a task is sometimes said to have a "disposition" of succeeded/failed/aborted. Another one is:
I can put five employees at your disposal.
This means that I can bring five employees to the appropriate place, where they will be ready to do as you command.
And finally, here's a stock phrase which is worth knowing, because it sheds a lot of light on the other senses: propose and dispose. Here's an example:
Men propose; women dispose. [Source]
This means that men offer possibilities to consider (propose), and women decide which proposals to carry out and which to reject (dispose—that is, sort out the proposals and bring some but not all to action).
Best Answer
OED 1 gives this definition:
The citations given here suggest that the term became current about 1817. OED offers no speculation about why the term was adopted. My guess is that it reflects the very colorful liveries which were popular in the 19th century. Livery is the formal uniform which wealthy people used to require their servants to wear. Livery was always markedly different from ordinary dress—you don't want your servants to be mistaken for your peers! As a form of 'conspicuous consumption' it was usually patterned on the formal dress of an earlier period and executed in bright, eye-catching colors, which contrasted particularly in the 19th century with the sober colors preferred by gentlemen. Here is an example of the livery worn by Queen Victoria's coachmen; note the powerful stripe effect:
@Stephie notes a source which reports that “The tiger wore an orange and black striped waistcoat which is what gave them their ‘title’.” It may very well be that the first footboy to be called a tiger wore such a waistcoat, and even that such a waistcoat was at some point the 'default' dress for footboys, just as green-and-black stripes were common for table-servants in the early 20th century. But I have been unable to find a contemporary illustration. It should also be noted that the ancient practice was that liveries were executed in colors specific to the family, so it cannot have been a universal practice.