When the first such day was held, "coming out", even for white lesbians and gay men and lesbians who were secure in their jobs and lives, was a risky proposition. The so-called "Gay Liberation" of the 1970's boomeranged into a conservative backlash during the 1980s. (I apologize for the mixed metaphors.)
One reason was the election of Ronald Reagan in 1980. Of course, the election didn't happen in a vacuum: There was a counter-revolution to the Gay Rights movement, just as there was for the Women's and Civil Rights movements. That counter-revolution was not always visible or audible: Most of the time, it was more like a series of tremors that one only felt when one happened to be in its path. Those tremors culminated in the earthquake of Reagan's election.
Perhaps even more important, though, was the outbreak of HIV/AIDs. In the early years of the epidemic, all of its known (or, at any rate, recorded) victims were believed to be gay men, intravenous drug users, Haitians or West Africans. The last three groups I've mentioned were near the bottom of the American socio-economic ladder; the early path of the epidemic caused many Americans to lump gay men with them. Naturally, that only helped to inflame existing prejudices against gay men.
Another reason, I believe, why "coming out" was difficult during the 1980's was that many people associated lesbians with the most shrill and hateful kinds of feminists (or pseudo-scholars who called themselves feminists, anyway). The conservatives and religious hatemongers who were spouting anti-gay rhetoric tended to look none too kindly on feminists anyway; the association those conservatives made between feminists and lesbians surely made things worse for both.
If it was difficult for gays and lesbians to come out in 1988, you can only imagine how much worse things were for trans people. Of the trans people I've met (which include everything from those who haven't yet begun-- or who have chosen not to-- to transition, to post-ops), it seems that there are, chronologically, two groups: the ones who transitioned during or before the early 1980's, and the ones who transitioned during or after the early 1990's.
If my observations in any way reflect what has happened throughout the trans community, there is a "lost generation" of trans people--the ones who didn't transition during the decade or so between the two groups I've mentioned. That period almost perfectly coincides with the conservative backlash I've mentioned against gays and lesbians, and that "lost generation" includes many who took their own lives or who died slower deaths from drug and alcohol abuse, as well as those who simply didn't transition and those--including yours truly--who transitioned later in their lives than they might have otherwise.
So, even though we have a long way to go, things are certainly better for us, in many ways, than they were in 1988. National Coming Out Day is one reason for that.