Latin and ancient Greek are both inflected languages—that is, the endings of words change to give us information about them. For example:
Aemilia sedet. Aemilia sits.
filia Aemiliae sedet. The daughter of Aemilia sits.
In the former example, Aemilia has the ending -a, telling us she is performing the sentence's action (sitting). In the latter, Aemilia has the ending -ae, telling us she is possessing something (in this case, a daughter).
Although English many centuries ago was also an inflected language, over time it has evolved into an analytic language.
The meaning which Latin conveys using case endings, English conveys using a combination of word order and 'helper' words (such as 'of' in the example above).
This makes things challenging for English-speaking learners of any inflected language. After all, we're not used to paying attention to the endings of words! But there are a few examples of inflection in English today which hark back to its inflected roots.
Amelia sees a girl. She sees a girl.
A girl sees Aemilia. The girl sees her.
Notice that when Amelia is doing the action (seeing) she can be replaced with the pronoun she. When the action is being done to her, Amelia is replaced by with the pronoun her. Likewise he becomes him, they becomes them, and we follow this pattern with neopronouns such as ey and em too.
We can see that English is becoming even less inflected: it is increasingly uncommon to say whom rather than who in everyday conversation. Who becoming whom is another example of the ending changing to show us whether it's the subject or the object.
One cat, many cats. One dog, many dogs. One mouse, many mice. Grammatical number is one way nouns can inflect. English may have dropped its cases, but nouns still change their ending to give us information about their number. (Languages like Mandarin and Japanese do without grammatical number altogether!)
Notice that apostrophe-s expresses possession, just like the genitive ending -ae in Latin. This is because apostrophe-s is a genitive ending hanging on from a much older form of English! Speakers of German, which still has a case system, will recognise the same genitive ending.
Though language has changed over time to make case endings and a lack of 'helper' words quite a difficult concept for English speakers, inflection becomes much easier to read with time and practice.
How to learn case endings
Little by little: there are many noun endings in Latin, too many to learn all at once! Try breaking them down and revising case by case, or declension by declension.
Look for patterns: for example, accusative singulars tend to end in -m. Every case has an interesting feature tying the declensions together.
Experiment with learning methods: online testers, creating your own table, colour-coding, and many more. There is no single right way to learn, so experiment to discover what's best for you.
And most importantly, come back to your endings regularly! Short, regular practices are the key to memorising any information you might need.