父はデジタルカメラの講習を受けに行ったが、あまりの難しさにびっくりしたようだ。
So this sentence is saying, "my dad went to take a short course on digital cameras, but it was difficult to the point of surprising him." Could you say
父はデジタルカメラの講習を受けに行ったが、あまりの難しさにびっくりした。
instead or would omitting the ようだ make the sentence incorrect. What is the point of the ようだ at the end of the sentence?
The sentence is complete and grammatically correct without ようだ. (Please also see Tsuyoshi Ito's answer.)
…ようだ adds "It seems that ...".
In this case,
父はデジタルカメラの講習を受けに行ったが、あまりの難しさにびっくりしたようだ。
My dad went to take a short course on digital cameras, but it seems that he was surprised at just how difficult it was.
父はデジタルカメラの講習を受けに行ったが、あまりの難しさにびっくりした。
This sentence is correct only in limited contexts.
It states the definite fact that the father got surprised. As Derek Schaab stated in this answer accurately,
Japanese has a curious unwritten rule which states, in essence, that you cannot presume to know the intimate details of a third person's mental state.
Therefore, if the author of the sentence is the father and using 父 as a first-person pronoun, then the sentence is fine, but otherwise you need ようだ.
By the way, for some reason, the sentence becomes natural if it ends with びっくりしていた instead of びっくりした when 父 refers to the author’s father. I do not know why.
This 〜ようだ means "it seems", and means you are making the statement as conjecture based on some kind of information. The first sentence says "It seems he was surprised by how difficult the course was". You might say this after seeing him confusingly tinkering around with the camera even after the course. If you leave off 〜ようだ, it's saying that he was definitely surprised by how difficult it was; maybe he told you, or your mother told you he said that, etc.
