The closest intuitive meaning of the f1-score is being perceived as the mean of the recall and the precision. Let's clear it for you :
In a classification task, you may be planning to build a classifier with high precision AND recall. For example, a classifier that tells if a person is honest or not.
For precision, you are able to usually tell accurately how many honest people out there in a given group. In this case, when caring about high precision, you assume that you can misclassify a liar person as honest but not often. In other words, here you are trying to identify liar from honest as a whole group.
However, for recall, you will be really concerned if you think a liar person to be honest. For you, this will be a great loss and a big mistake and you don't want to do it again. Also, it's okay if you classified someone honest as a liar but your model should never (or mostly not to) claim a liar person as honest. In other words, here you are focusing on a specific class and you are trying not to make a mistake about it.
Now, let take the case where you want your model to (1) precisely identify honest from a liar (precision) (2) identify each person from both classes (recall). Which means that you will select the model that will perform well on both metric.
You model selection decision will then try to evaluate each model based on the mean of the two metrics. F-Score is the best one that can describe this. Let's have a look on the formula:
Recall: p = tp/(tp+fp)
Recall: r = tp/(tp+fn)
F-score: fscore = 2/(1/r+1/p)
As you see, the higher recall AND precision, the higher F-score.