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It's always difficult to answer questions as to 'why' a certain grammatical pattern is used in different languages. In Spanish, in the sentences you have used as examples for your question, the answer is simply that the preposition 'de' must come before the infinitive of the verb, having a meaning of 'to' or 'of'/from', respectively.
"Estoy cansado de trabajar" means "I'm tired of working." or "I'm tired from working."
In the first sentence, "Estoy feliz de conocerte," 'de' must come between the adjective 'feliz' and the infinitive 'conocerte,' in order for the sentence to be grammatically correct.
English grammar is quite different than Spanish grammar and rules will vary. Please see the following webpage for a more in depth explanation of using 'de' in different circumstances: CLICK
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