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They're not. For many conservatives, their fear lies in the creeping overt indoctrination practices so many in the "pro-trans" community seem to practice. 
For many, it feels so much like the dishonesty and lies they suffered under during the same-sex marriage movement.
We were promised honesty, but was it HONESTY to force some kids to miss out on having their Dad?
We were told the homosexual community owned the franchise on what love really means, but is it LOVING to destroy the primal love that exists between a baby and its Mother?
The homosexual community insisted THEY were the tolerant ones, but is it TOLERANT to silence others with "anti-discrimination" laws? (Gay people have the right to live as they choose, but no right to silence others who disagree with them on marriage and sexual right and wrong.)
Homosexuals insist they know what's best when it comes to informing others, but is it RIGHT to force homosexual education on ALL children? Do they allow equal time to opposing view points to be taught to their children?
The homosexual community insisted THEY were the honest ones when campaigning for their changes, but is it HONEST to lie about what you plan to do regarding marriage. ("Fighting for gay marriage generally involves LYING about what we are going to do with marriage when we get there, because we lie that the institution of marriage is not going to change...and this is a LIE. The institution of marriage is going to change, and it should change, and again I don't think it
should exist." - Masha Gessen, Lesbian Activist")
So is any of this EVEN NECESSARY now that all couples have equal benefits? (The truth is, from GAY COUPLES to the most "hyper-sectional Trans-Queer" person you can find, each of them have EQUAL BENEFITS under the law.)
HOWEVER - The relationship of couples to a child of any age is a different question COMPLETELY. In some states, same-sex couples do not have the right to adopt or create a child by surrogacy - because the right of the child to have both a mother and father role model outweighs the claims of same-sex adults to obtain a child.
Same-sex "marriage" abolishes either a mother or a father from the life of a child. It denies EQUALITY to children who become the victim, by design, of adults more eager to feed their selfish wishes than satisfy the most critical needs of an innocent child.
What so many in the trans community fail to acknowledge is, "the world doesn't owe them understanding". They're owed equal PROTECTION under the law, but not equal outcomes. 
To answer your question more precisely, most conservatives don't "fear" people in the trans community, nor do they empathy towards whatever personal struggles they might be dealing with. Perhaps you're making some of the same misjudgments toward conservatives you accuse them of making toward members of the trans community? (If you were in court, you might hear, "Objection your honor. Attempting to argue facts not in evidence." Personally, I can't for a moment imagine anyone from the trans community being rejected or looked down upon for attempting to enter my local Catholic church.
Conservatives don't fear anyone in the trans community. Instead, they fear being lied to in the same way they were during the far left's campaign to undermine the institution of traditional marriage. In the same way the homosexual community declared in the beginning they only wanted a chance to share in an equivalent legal bond with their partner, then it was private unions, then same sex marriage, etc...conservatives know what started as "we only want to be free to be who we really are" will soon turn into a demand for unearned privileges at the expense of other people's liberty.
Why is the far right so deathly afraid of trans people? Could it be that they're incapable of empathy or understanding and hate or fear whatever they don't understand?