I'm going to present an excerpt from the excellent book Economics in One Lesson by Hazlitt, which is available free online from the Mises institute.

An American manufacturer of woolen sweaters goes to Congress
or to the State Department and tells the committee or officials concerned
that it would be a national disaster for them to remove or
reduce the tariff on British sweaters. He now sells his sweaters for
$15 each, but English manufacturers could sell here sweaters of the
same quality for $10. A duty of $5, therefore, is needed to keep him in business. He is not thinking of himself, of course, but of the thousand
men and women he employs, and of the people to whom their
spending in turn gives employment. Throw them out of work, and
you create unemployment and a fall in purchasing power, which would
spread in ever-widening circles. And if he can prove that he really
would be forced out of business if the tariff were removed or
reduced, his argument against that action is regarded by Congress as
conclusive.
But the fallacy comes from looking merely at this manufacturer
and his employees, or merely at the American sweater industry. It
comes from noticing only the results that are immediately seen, and
neglecting the results that are not seen because they are prevented
from coming into existence.
The lobbyists for tariff protection are continually putting forward
arguments that are not factually correct. But let us assume that the
facts in this case are precisely as the sweater manufacturer has stated
them. Let us assume that a tariff of $5 a sweater is necessary for him
to stay in business and provide employment at sweater making for his
workers.
We have deliberately chosen the most unfavorable example of any
for the removal of a tariff. We have not taken an argument for the
imposition of a new tariff in order to bring a new industry into existence,
but an argument for the retention of a tariff that has already
brought an industry into existence, and cannot be repealed without hurting
somebody.
The tariff is repealed; the manufacturer goes out of business; a
thousand workers are laid off; the particular tradesmen whom they
patronized are hurt. This is the immediate result that is seen. But there
are also results which, while much more difficult to trace, are no less
immediate and no less real. For now sweaters that formerly cost $15
apiece can be bought for $10. Consumers can now buy the same quality
of sweater for less money, or a much better one for the same
money. If they buy the same quality of sweater, they not only get the
sweater, but they have $5 left over, which they would not have had
under the previous conditions, to buy something else. With the $5 left over they help employment in any number
of other industries in the United States

Now let us look at the matter the other way round, and see the effect
of imposing a tariff in the first place. Suppose that there had been no tariff
on foreign knit goods, that Americans were accustomed to buying foreign
sweaters without duty, and that the argument were then put forward
that we could bring a sweater industry into existence by imposing a duty of $5 on
sweaters.
There would be nothing logically wrong with this argument so far
as it went. The cost of British sweaters to the American consumermight thereby be forced so high that American manufacturers would
find it profitable to enter the sweater business. But American consumers
would be forced to subsidize this industry. On every American
sweater they bought they would be forced in effect to pay a tax of $5
which would be collected from them in a higher price by the new
sweater industry.

In short tariffs are a form of tax, we just don't notice it right away.