Subjective facts as-in those with different sources or ways of measuring information. For example, people can choose the scientific studies that they draw attention to, and find an expert that agrees with them. (Also, the way in which people use the term 'fact' often relates to empirical observation and this can be miscalibrated: what was considered objective fact sometimes gets disproved or constrained.)

Alternative facts as-in alternative selections of facts. Not merely a case of adding facts, but also in not acknowledging the facts that you don't like.

Aside: @smackay's comment is very close in meaning to what I wrote (https://news.ycombinator.com/item?id=13480772). You might find it easier to understand.

> The indiscriminate usage of the term by the mainstream media is a strategic mistake since they are giving ground to the frame of all facts being subjective and equally applicable

They're saying that the media using the term "alternative facts" is giving ground to the very people who initially began spreading the lies, and in doing so is giving them an opening to exploit. Those people want to portray facts as subjective and all equally applicable. They aren't, and we should expect the the same people who started spouting "alternative facts" (read: lies) to begin branding actual facts as "alternative facts".