Russell's theory of descriptions was his attempt to apply the tools of logical analysis to problems in the philosophy of language. Russell's theory is what is called a mediated reference theory, or sometimes an indirect reference theory. This is because in Russell's theory, we do not directly reference things. Instead, we describe things in a specific way such that the things we want to reference are uniquely satisfied. This leads to the very confusing and unnatural sounding sentences such as those in your example, but it is easy to following along with his examples and understand how they mean the same thing as the more natural sounding English expressions they replace.
Why Russell advocated such a seemingly unnatural position is because of a series of puzzles discovered by Frege about language that caused Frege and others (such as Russell) to begin the modern study of the philosophy of language. One puzzle in the philosophy of language is "how can we reference things that do not exist?" commonly called "the problem of apparent reference to non-existentials" and Russell's famous example is:
The present king of France is bald.
France has no present king, therefore the sentence cannot be true. However, it wouldn't seem right to say that it is false either, because that would mean "the present king of France is not bald" is true, which it also is not. So, how can this sentence be meaningful (how can we understand it?) if it doesn't reference anything? Russell uses his theory of descriptions to analyze the sentence as really expressing a conjunction of three sentences:
(a) There exists something that is the present king of France.
(b) There is only one thing that is the present king of France.
(c) Anything that is the present king of France is bald.
Clearly we can see that (a) is false, so no matter what the other sentences say, the entire conjunction (a AND b AND c) is false. The sentence is meaningful because it presents us with a false statement, one that is logically analyzable, as opposed to some weird and incoherent string of symbols. This should also highlight how this is an "indirect" theory of reference. We are not saying "he is the king of France" while pointing at someone. We are saying "something or other is the king of France" and "whatever that something is is bald". We never pick out that one individual thing so we are not referencing something that does not exist.
In the most straightforward way possible, to Russell the sentence "I have two hands" means:
There exists an object "a".
There exists an object "b".
"a" and "b" are not identical, which means they have some properties that are different.
The property, or the predicate, "x is a hand of mine" is true only when we substitute "a" or "b" in for "x".
As a quick aside, if you are confused about the word "substitute", treat it exactly like in algebra class. "5+x=10" is true only when we substitute "5" in for "x". It is not true when we substitute in "6", "-12", or any other number.
What this sentence means is that "a" and "b" are the only things in the world, out of every object that exists, that will make the predicate "x is a hand of mine" true; therefore, they are your hands. The fact that they are not identical assures us that there is more than one, but as Russell said, it is never stated that there are two of them, only that they are not identical and that they are the only things that make the predicate true. No other objects in the world make all of those statements true at the same time besides your hands, so that conjunction of sentences uniquely picks out your hands. It is only after the fact that we can perceive that there are only two objects.
The same goes for the satellite example:
There exists an object "c"
The predicate "x is a satellite of Earth" is true only when we substitute "c" in for "x".
Russell wrote this example long before we had the ability to send objects into space and into orbit around the Earth. So, interestingly enough, his satellite example we can now see is clearly false. At the time of his writing, the Moon was the only Earth satellite that we knew of. Today, however, the Earth has many satellites, including the International Space Station, for example. If someone were to assert those two sentences today, the first sentence is true, however the second is not. This is because there are other objects, such as the ISS, that satisfy the predicate. So, while that sentence might have been true during Russell's time, it is not true now. However, whether it is true or false, it is one or the other so it is still meaningful in Russell's theory.
So, ultimately how Russell's analyses of descriptions purport to mean the same thing as normal English sentences is by trying to explain that the simple sentences we say actually mean more complex things. If they did not, we would continuously run into Frege's puzzles and we would not have any sort of theory of meaning. It is an empirical fact that language does have meaning, so we know that whatever theory explains the meaning has to answer Frege's Puzzles. That being said, Russell's descriptions are not the only proposed theory of reference or meaning. Saul Kripke's Naming & Necessity is the most famous criticism of Frege and Russell's theory of reference (they had different theories but he critiques them both) in regards to proper names, at least. I'd recommend starting with the IEP's article on philosophy of language if you want to understand more about how this problem arose in the study of logic and mathematics and how the field has developed since then. I'd recommend Lycan's Philosophy of Language: A Contemporary Introduction for a very approachable introduction to the field and Soames' Philosophy of Language for a more advanced introduction.