Translation:The roe deer's fawn is running there in the forest.
I highly doubt that these are useful words to learn in an early phase of language learning. I have no idea what a fawn is, and this is true for nearly all other baby animals. I speak four languages fluently, but only in my native language I would be able to recall that word, with difficulty. I don't know if it is because of my lack of interest in wild animals, but I have no idea why I should know this in Hungarian. The same is true for animal sounds. In Dutch which I use on everyday basis, I do not know any of those special words except for 'barking' and I'm fine. Therefore I believe that it is useless vocabulary on this level.
In 10 visits totalling about 15 months, I think I have actually seen more őz in Hungary than macska (about a dozen compared to one). Not that many ducks either - geese and hens but not ducks. The only szarvasok I have seen there were on a hunting reservation. No bears or wolves.
Roe deer (Capreolus capreolus; őz) are not female deer (Cervus spp.; szarvas), though apparently some Hungarians think so. (As do some Germans, for that matter.)
The size (roe deer are smaller than "real" deer) may be an influence?
Roe deer are fairly common in Europe but don't seem to live elsewhere; perhaps that is why you are not familiar with them.
As an 'animal factoid master', you will want to know this about 'deer':
1. The smaller European animal is known as the 'roe deer' or 'roe' (German 'Reh')
2. The larger animal is known as the 'red deer' or 'hart' (German 'Hirsch')
3. The North American animal usually called 'deer' is the 'white-tailed deer' or 'whitetail'. This species does not live in Europe.
Even in America, 'Hirsch' is sometimes seen as a family name (e.g., Judd Hirsch).
The original name of the American retail store 'Sears' was Sears, Roebuck and Co. The name 'Roebuck' refers to a male roe deer.