If a person gets engaged to a woman using terumot, terumot are the produce of Eretz Yisrael (Land of Israel) that have to be separated and designated aside to be given to a Kohen (priest).
If a Kohen, for example, uses them to get engaged to a woman, or ma'asrot, a broad term to refer to all different types of gifts that are given to the Levi (Levite), or, in situations, others, if a person uses that produce that's supposed to go to the Levi, or that's went to a Levi, in order to affect kiddushin (engaged halachically)—all different gifts that are usually given to Kohanim, different parts of the animal, or spring water that's usually mixed with the ashes of the Parah Adumah (red heifer), or ashes of the Parah Adumah—in all of these cases, the woman is engaged halachically.
When it comes to these items, even if the person has them and is a Kohen, they have some value to him; they're his, and he could use them to betroth the woman, even if the woman is not a Kohen. He gives it to her—the terumah—and she can then use that terumah and sell it back to another Kohen, so it has value to her. The same is true of the ma'asrot and the matanot (gifts).
When it comes to Mei Chatat and Efer Chatat—the water that's mixed with the Parah Adumah, ashes, and the ashes themselves—they also have value because, even though the items themselves belong to the Bet HaMikdash (Temple), the woman now has them, and she could have a service where she brings them to the people who need them, who need to be purified, and she could charge for that service. So she has something of value; she has a vehicle to make money with. With these items, she could charge a delivery fee, and she has monetary benefits, so by giving them to her, you are now engaged to her halachically.
And even if Yisrael, someone who's not a Kohen or a Levi, they don't usually use terumah or matanot or ma'asrot and all that, they can use these items if they happen to own them. Let's say they got them as an inheritance somehow. If a Kohen died and he didn't have any sons, and he left it to his daughter, even though his daughter was married to Yisrael, she would now inherit that terumah. So when she died, her husband, who wasn't a Kohen but was Yisrael, takes that and inherits the terumah from her. Therefore, he can use it in order to marry another woman.