“Moses’ anger burned.” An idiom for the fact that Moses became very angry. When someone gets very angry their skin flushes and they get physically hot.
“and he threw the tablets out of his hands.” This record of Moses is very human and gives us an example of how human emotion often works. God had already told Moses that the people had made a gold calf god and were worshiping it (Exod. 32:7-8), so Moses knew intellectually what the people were doing. But there is a big difference between “head knowledge” and actually experiencing something. In this case, although Moses knew the people had made a gold calf god and were worshiping it, when he actually saw what the people were doing he was filled with emotion and threw down the stone tablets God had made and broke them. In the same way, most everyone has examples of times in life when we know about something that is bad or evil and yet can “keep it together” and not become overly emotional, but then when we come face to face with the bad thing we are overcome with emotion. That is a typically human trait.
We can understand why Moses would feel the way he did and break the tablets. The very first two commandments were “I am Yahweh your God...You must not have any other gods besides me. You must not make for yourself a carved image...you must not bow down to them and you must not serve them….” (Exod. 20:2-5). The people of Israel agreed to those commandments and made a covenant with God that they would obey them, and got sprinkled with covenant blood (Exod. 24:7-8). Furthermore, God had shown the leaders of Israel that He was on the mountain with Moses (Exod. 24:9-11). But now, after only a month or so of Moses being gone the Israelites completely abandoned the commitment they had made to God and turned to idols (Moses was gone for 40 days, but it would have taken a while to convince Aaron to make the calf god and then to actually make it and begin worship ceremonies, so it was likely a month or even less that Moses was gone when the process started).
There is also strong evidence that sexual activity that God would never tolerate was a prime motivator in the people making a calf god, which means that the people not only ignored the commandments they had agreed to about not making idols, they also ignored God’s commands about sexual purity (see the commentary on Exod. 32:6). In any case, when Moses actually saw for himself what the people were doing and realized that the people had broken many of the commandments that they had agreed to and some of which were written on the very tablets Moses was carrying, he was furious and broke the tablets because at that moment it must have seemed to Moses that the covenant was pointless.
“at the foot of the mountain.” The Hebrew text literally reads, “under” or “beneath” the mountain, but it means “at the foot of.” Saying “under the mountain” would be unclear in English. Moses broke the tablets with the Ten Commandments representing the covenant Israel made with God in the same place that Israel had made that covenant with God some 40 days earlier (Exod. 24:4).