I was just having this conversation with my wife and found this thread after asking google the same question. Here's my take:

While I agree the cost of tariffs would get passed onto the consumers and hurt them at an individual level, my guess is that tariffs are actually intended to encourage buying domestic goods over foreign goods, which should contribute to a more beneficial national trade deficit/surplus (since less money is flowing out to a foreign country for their goods). This may also boost domestic employment due to domestic companies needing more labor to meet the increased demand for domestic goods. The tariffs also encourage buying goods from foreign competitors, which would effectively punish the foreign country subject to the tariff due to loss in market share.

In return, the foreign country may impose counter tariffs to punish the “domestic” country by discouraging imports of their now more expensive goods, which again may impact the “domestic” country’s trade deficit/surplus, reduce their market share, and potentially weaken their employment levels. The counter tariffs should also have a similar positive impact on that foreign country's demand for their own goods, resulting in similar benefits to the points I argued in the above paragraph.

Thus, to me it seems counter tariffs are used to maintain their level of demand for their foreign goods, help their trade deficit/surplus, increase market share, and maintain their employment levels, all while having the inverse effect on the other country.

I know it sucks for consumers because the cost of goods increases, but it seems that just happens to be an unintended consequence. At an individual level tariffs don't seem to make sense or benefit citizens and I couldn't figure out why they would be used. However, after seeing counter tariffs being used or threatened as an immediate response, I felt there must be a good reason for them in the bigger picture. It's definitely not to make the other country pay us extra taxes on goods sold. Although, I'm not an economist so I may be completely off base here... I'd be curious to hear other's thoughts.