Well, I'm not FeatheryFlorence, but am fluent in several languages and speak and understand quite a few more.
As you learn more, you start to get a real knack for spotting patterns, learning patterns, getting to know the sound changes that occur across words in related languages and within languages, and extrapolating or hypothesising about the likely meanings of words that you come across.
And this gets easier and easier the more you learn. It's also a whole lot of fun. Sort of like doing crossword puzzles!
Some examples of the process, using words that are already in English.
You already know the Japanese word karate, right?
So now you learn that it is made up of two parts, each of which have a meaning:
Kara means empty
Te means hand or hands
Now you come across the word karaoke, so you hypothesise that the word kara has the same meaning of 'empty', so now you only have to find out the meaning of 'oke'. Which turns out to be the chopped down form of the loan word 'orchestra', so literally 'empty orchestra' as in the music is provided, but it's empty of the vocals because you, the singer, provides them.
Now let's take the word origami.
Again two parts:
Ori from the verb oru, meaning to fold
Gami from the word kami (yes, a sound change occured), meaning paper
So origami literally means 'folding paper'.
And now you come across a word you don't know from English: tegami
So what does it mean? You can work out the separate parts
Te
Gami
And you work from there.
And I'm going to be mean here and not give away the meaning!
Completely different language. Let's say you see the Spanish word 'bailar'.
Well, in English you already know the words 'ballet' and 'ballerina' so you immediately hypothesise that it probably means 'to dance', tuck that away in the back of your head to see if the next time you come across the word in a sentence your hypothesis is proven or disproven, and just keep on going.
So the hardest part is the earliest stages of your first foreign language. At that point you have less of a mental framework of how the language operates, as well as fewer vocabulary 'hooks' to hang the meaning onto.
That means that in the early stages you will probably have to simply rote memorise a whole bunch of things until you reach a stage where it all starts to become clearer. And the faster you try to get through that stage, the better it is, because that part is the least fun but really is just a stepping stone to reaching more understanding.