1. Something has always existed. (Premise)
The MTW and the traditional Third Way both use (1) as a starting point.
2. Temporally contingent beings exist. (Premise)
3. If there is no temporally necessary being, then only temporally contingent beings exist. (Premise)
4. If only temporally contingent beings exist, it is necessarily the case at all past times that at least one temporally contingent being existed. (From 1 and 3)
5. It is not necessary for there to be any temporally contingent beings. (Premise)
6. Therefore, a temporally necessary being exists. (From 3 - 5)
To repeat, premise (1) is supported by the fact that out of nothing comes nothing. (2) and (3) are uncontroversial, and (4) is logically deducted, so that leave us with (5).
To put it bluntly, a denial of (5) results in plainly weird consequences. How would the non-existence of temporally contingent beings explain the existence of some other temporally contingent being? Does the non-existence of every non-unicorn imply that a unicorn exists?  Clearly not. Yet, if it's necessary that something has always existed, but it's not necessary that there always existed some temporally contingent being, it follows that a temporally necessary being exists.
 Alexander Pruss, "Why Is There Something Rather Than Nothing," Philosophia Christi 7 (2005), p. 210.