Koyunların arkasında bir kuzu daha vardı; annem onu görünce çok sevindi.

Because Turkish usually expresses behind X, in front of X, on top of X, etc. with a possessive-style structure.

• koyunlar = sheep
• koyunların = of the sheep / the sheep’s

So koyunların arkasında literally means something like at the sheep’s back / behind the sheep.

This is very common in Turkish:

• evin önünde = in front of the house
• masanın üstünde = on the table
• koyunların arkasında = behind the sheep

It breaks down like this:

• arka = back, rear
• -sı = 3rd person possessive ending
• -nda = locative ending -da/-de, with buffer n

So:

• arkası = its back / the back of it
• arkasında = on/at/behind its back

Together with koyunların, it becomes koyunların arkasında = behind the sheep.

This pattern is extremely common with location words in Turkish:

• önünde = in front of
• arkasında = behind
• yanında = beside
• üstünde = on top of
• altında = under

It does both, depending on context.

In bir kuzu daha, bir can be understood as the indefinite article-like a, but it also helps the phrase mean one more.

So:

• bir kuzu = a lamb
• bir kuzu daha = another lamb / one more lamb

Turkish does not have a separate word exactly like English a/an, so bir often fills that role.

Here daha means more in the sense of another or one more.

So:

• bir kuzu daha = another lamb / one more lamb

This is different from comparative uses like:

• daha büyük = bigger
• daha güzel = more beautiful

In this sentence, it is not comparing anything. It is simply adding one more item.

Because the sentence is in the past.

• var = there is / exists
• vardı = there was / existed

So:

• bir kuzu daha var = there is another lamb
• bir kuzu daha vardı = there was another lamb

Turkish uses var and yok for existence or presence, rather than a normal verb like English to be.

Because anne takes the 1st person singular possessive ending -m:

• anne = mother
• annem = my mother

Turkish very often uses possessive endings instead of separate words like my.

You could also say:

• benim annem

But that is more explicit or emphatic. Usually annem by itself is enough.

onu is the accusative form of o.

• o = he / she / it / that
• onu = him / her / it / that one

Here it means it, referring to the lamb.

It is in the accusative because it is a specific direct object:

• my mother saw it

In Turkish, specific/definite direct objects often take the accusative:

• onu gördü = she saw it
• kuzuyu gördü = she saw the lamb

görünce means when (someone) sees/saw or upon seeing.

It comes from:

• görmek = to see
• stem: gör-
• -ünce = a converb ending meaning when, once, upon

So:

• onu görünce = when she saw it / upon seeing it

This form connects actions:

• Eve gelince ararım. = When I come home, I’ll call.
• Seni görünce sevindim. = I was happy when I saw you.

In your sentence, annem onu görünce çok sevindi means literally something like:

• My mother, when she saw it, became very happy

Not by itself. The subject is understood from context.

In annem onu görünce çok sevindi, the natural reading is:

• annem is the one who saw it
• annem is also the one who became happy

So the subject of görünce is understood to be the same as the subject of the main clause unless something else makes it clear.

sevindi comes from sevinmek, not sevmek.

• sevinmek = to be glad, to rejoice, to become happy
• sevindi = was glad / became happy / rejoiced

Compare:

• sevdi from sevmek = loved / liked
• sevindi from sevinmek = was happy / was delighted

So in this sentence, çok sevindi means:

• was very happy
• was delighted

not loved it very much.

Because Turkish normally puts the main finite verb at or near the end.

So:

• annem onu görünce çok sevindi

has this rough order:

• annem = my mother
• onu görünce = when she saw it
• çok = very
• sevindi = became happy / was glad

A literal-feeling English order would be:

• My mother, when she saw it, became very happy

Turkish word order is flexible, but verb-final structure is the default and most natural pattern in sentences like this.