Learn German: Ich probiere dieses Kleid an. - I am trying on this dress.

The word for dress is a neuter noun, das Kleid. Because it is the direct object in this sentence (the thing you are trying on), it is in the accusative case. For neuter nouns in the accusative case, the demonstrative pronoun for this takes an -es ending, becoming dieses.

Why is the word an at the very end of the sentence?

The full verb here is anprobieren (to try on). This is a separable verb. In a normal statement, the main part of the verb (probiere) goes into the second position, and the prefix (an) is pushed to the very end of the sentence.

Could I drop the an and just say Ich probiere dieses Kleid?

No, because probieren by itself usually means to taste food or to attempt an action. When talking about trying on clothes, you must use the separable verb anprobieren, so the an at the end is required.

The English translation says I am trying on, but there is no word for am in German. Why?

German does not have an equivalent to the English continuous tense (the -ing form). The simple present tense ich probiere translates to both I try and I am trying, depending on the context.