The verb werden is an interesting one. It can be both a full verb meaning to become or it can be an auxiliary verb used to indicate future tense.
In the first of your two examples, the meaning of the English verb is comparable to to become. Hence, one can use the full verb werden to phrase this in German.
It’s growing colder → Es wird kälter.
In the second of your examples, the machine translation used werden, the auxiliary, to indicate future tense much like the auxiliary will in English.
It will be colder → Es wird kälter sein.
The key here is that the machine translation — while undoubtedly generating correct German sentences — decided to render the English future tense differently in both examples. English is relatively strict when it comes to using future tense: if the topic of the sentence is something in the future, future tense should be used except in rare examples when present tense is permissive. French is even stricter and future tenses are always used when talking about the future (but in French the future tense is not an auxiliary-dependent tense but generated by inflection; if an inflected tense exists for a certain direction of time the language will probably require using it, as a very general rule).
In German, however, the aspect of future is typically encoded semantically and not grammatically. An adverbial of time signifying some point in the future (e.g. morgen, tomorrow), a verb that from its meaning alone already points into the future or any other semantic clue that we are talking about the future is sufficient to encode the information future — future tense is used rather rarely.
Let’s take another look at the two German sentences. Werden — the full verb — means to become i.e. it indicates a process that starts at a given point and ends at a different point. If used in present tense, the starting point could be anywhere (past, present or future) but the end point will always be in the future. Thus, simply by our choice of verb we have encoded the information future into our sentence and marking the future tense by using werden, the auxiliary, is not necessary (but permitted!). (If the end point were a point in the past or at present, past tense of werden, the full verb, would be used.)
Sein, however, indicates a status that is typically interpreted to be ongoing. Without any additional semantic information, the present tense of sein is indeed assumed to be present. Hence, if you only have the information present in the English original and want to preserve the information that you are speaking about the future, sein must be in future tense by using the auxiliary werden. However, if you decide to use any other information that encodes future (see example below) you can again skip back to present tense.
Tomorrow it will be colder. → Morgen ist es kälter. (less common)
Tomorrow it will snow. → Morgen schneit es. (more common)
(During the process of writing this up, I realised what a good job the translation machine did with these sentences.)
I mentioned above that encoding future by tense is unnecessary when dealing with something semantic that indicates future. It is, however, possible and sometimes used to emphasise the future aspect. For example, it is not wrong (and even sufficiently common) to use werden in future tense in the first sentence:
Es wird kälter werden.
