Did Fidel Castro put gay people into concentration camps?
This claim is a fact if you consider "concentration camps" to be metaphoric.
The camps were called UMAP:
https://en.wikipedia.org/wiki/Military_Units_to_Aid_Production
They were agricultural (forced) labor camps operated by the Cuban government.
This period is known in Cuban history as "Quinquenio Gris" (The Gray Quinquennium). In recent years the Cuban government has discretely started to acknowledge their atrocities during this period.
Here is government-own Wikipedia-like page about the topic in Spanish (watered-down, with lots of added sugar):
https://www.ecured.cu/Quinquenio_gris
Here is a blog-post (in English) from a group of bloggers that support the Cuban government (although they have been censored every now and then by the very same government they claim to support):
http://cuba.blogspot.ca/2007/02/gray-quinquennium-or-five-black-decades.html
Speeches from Fidel Castro that gave birth to the nightmare (also in Spanish):
http://www.cuba.cu/gobierno/discursos/1961/esp/f300661e.html
http://www.cuba.cu/gobierno/discursos/1963/esp/f130363e.html
In the second one he said:
(...) Muchos de esos pepillos vagos, hijos de burgueses, andan por ahí
con unos pantaloncitos demasiado estrechos (RISAS); algunos de ellos
con una guitarrita en actitudes “elvispreslianas”, y que han llevado
su libertinaje a extremos de querer ir a algunos sitios de
concurrencia pública a organizar sus shows feminoides por la libre.
Que no confundan la serenidad de la Revolución y la ecuanimidad de la
Revolución con debilidades de la Revolución. Porque nuestra sociedad
no puede darles cabida a esas degeneraciones (APLAUSOS). La sociedad
socialista no puede permitir ese tipo de degeneraciones. ¿Jovencitos
aspirantes a eso? ¡No! “Arbol que creció torcido...”, ya el remedio
no es tan fácil. No voy a decir que vayamos a aplicar medidas
drásticas contra esos árboles torcidos, pero jovencitos aspirantes,
¡no! Hay unas cuantas teorías, yo no soy científico, no soy un
técnico en esa materia (RISAS), pero sí observé siempre una cosa: que
el campo no daba ese subproducto. Siempre observé eso, y siempre lo
tengo muy presente.
Summary: He is saying that his Revolution would not accept such kind of "feebleness", such kind of "degeneration". He said that even if he is not a scientist, he observed that agricultural regions never have such kind of "by products".
The first speech gave ground for sending to the UMAPs many writers, , professors, musicians and actors who performed, wrote or filmed something that the government did not like.
With the second speech, hell broke loose, the witch hunting begun, and it lasted several years.
The claim about Democrats praising F.Castro is... well, anecdotal at best. In general terms, it is ridiculous to say that Democrats looked up to him in any way. They were responsible for most of the hardest actions taken against his government over the course of history: