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For example, Latin is a source language for new words in English and other European languages, and I know English, Sanskrit and Arabic are also source languages in many other languages.
What are the reasons that languages get picked for that? What are the examples? From the examples above it looks like religion is a strong reason (Latin, Sanskrit, Arabic).
I think that the reasons can all come under the umbrella of sociological reasons; that is, purely linguistic reasons are rare.
Religion (Arabic vocab in Farsi, Urdu, Turkish because of Islam) or social status (Greek vocab in arts in Latin) or prestige (Norman vocab in law making in English) or education status (English for electronics on other languages) or technology (Dutch sailing and shipbuilding terms in English) or lingua franca (Swahili vocab in East Africa) are all sociological reasons.
There might be linguistic reasons for frequency of borrowings. I could conjecture that borrowings might occur more likely from a nearby dialect than from a far off unrelated language. But it just seems that once a dialect becomes unintelligible, it might as well be from Mars, and that linguistic similarity just doesn't translate at all to real world, folk similarity (e.g. English 'friendship' and German 'Freundschaft' are etymologically very cognate, but not recognizable informally, and that similarity or lack thereof would not encourage or discourage borrowing).
Loanwords, or borrowings, are a consequence of two languages having a contact. When there is an asymmetry between the two languages, a language takes the missing term from the other word, and sometimes the borrowing is not just for lexicon, but also syntax, etc.
The borrowings can go in both directions, so what makes one of those two a source language? The prestige and the importance of a language over the other one is perhaps the main reason.
Latin, for example, became a Lingua Franca because through the expansion of the Roman Empire it acquired importance and prestige, being the official language of the Empire.
Then, a lot of terms entered the various european languages, because Latin, after the fall of the Western Roman Empire, was the Lingua Franca for over 1000 years in Europe in many fields: culture, science, international relationships. Because of this it affected all the languages it came in contact with, even when it started to be used less, replaced by English.
Nowadays, English is the Lingua Franca, not just in Europe but worldwide (at least in most media). Because of this most speakers know some English enough to be able to take some words from it. For a deep explanation and also examples of loanwords from the various languages, visit this page.
Reasons for borrowing
Words are usually borrowed from a language of a former colonizing power. For example, in Kyrgyz many words borrowed from Russian, so in England - from the language of Roman Empire.
