The dramatic expansion of Yamna is related to the introducing of food producing in Steppe

 The dramatic expansion of Yamna is related to the introducing of food producing in Steppe. Suddenly large grasslands became pastures for herding, which itself resulted in dramatic increase of population.

 Farming was invented in Near East , Fertile Crescent. See in maps. This period is known as Neolithic and starts at 9500 BC. The oldest evidence of crop raising is found in Diyarbekir region of Turkey. Farmers moved to different directions. They had diverse haplogroups and almost certainly were speaking different languages.

 Levantine farmers moved to Africa. Farmers from Anatolia moved to Europe. Iranian farmers moved to Central Asia and India ( this latter is disputed ) . And farmers from historic Armenia moved to Caucasus and then farming in the form of pastoralism appear in Steppe, North of Caucasus.

For IE puzzle there are two main options. 

1. PIE was the language of farmers living in historic Armenia who moved to North mixed with locales and expanded there as Yamna.

2. PIE was the language of Steppic hunters who learned farming (borrowed terminology) and metallurgy from Near Eastern farmers. And then expanded. 

There is a third option also that farmers of west Ukraine who themselves were from Anatolia affected Steppe. But this option is not anymore popular after the paleogenetics findings.

In both cases we would have Near Eastern terminology in IE. Some examples.

gud in Sumerian means ox, bull and we have kov, cow in IE languages.

hawirni means lamb in Hurrian, Hovi in PIE sheep

tawr in Semitic and IE languages related to cattle.

urud in Sumerian means copper, and hreud is the red color in PIE.

Also numbers like six, seven and word wine but this latter probably is from IE to Kartvelian