In most cases, the present continuous tense of a verb is appending "ing" to the end of the verb, like "playing" for "play", but why does "mimic" become "mimicking"? That's so strange.
It's a standard spelling change. Verbs ending in c form their present particle as -cking. This helps the pronunciation, as c at the end of a word is normally "hard" (pronounced like k) but when followed by an "i" it becomes soft (pronounced like s). But the pronunciation of mimicking has a hard sound. Inserting a "k" helps to show that.
So all the verbs "frolic", "panic" "picnic" and "bivouac" all change to "ck". THe only exception I can think of is "arcing".
The answer is about pronunciation, but it's not just about hard "c" vs soft "c".
Many words double the final consonant before adding -ing, such as "bid" -> "bidding". The reason for this is to avoid confusion with a so-called "magic e" being omitted, as in "bide" -> "biding". That is, the doubling indicates that the previous vowel is short. Some form of this is necessary for "mimic", because a word ending -icing would be pronounced with a long vowel, as in "slicing".
Now it is a peculiarity of "c" that if you want to double a hard "c", you (almost always) instead replace the second "c" by a "k". This is because "cc" is normally pronounced differently, as in "accept".
This also explains why "syncing", "arcing", etc., don't add the k. Because there are two consonants before -ing, you do not need to double anything to get a short vowel.
When forming derivative forms of words in English, pronunciation is almost always preserved and the spelling is modified if needed to indicate the correct pronunciation.
In normal English spelling, a ‘c’ either in a word-initial position or proceeded by a vowel is pronounced as an /s/ instead of a /k/ if it is also followed by an ‘e’, ‘i’, or a ‘y’. For example the syllables ‘cin’ and ‘sin’ have identical initial consonants in English.
The final ‘c’ in ‘mimic’ fits this second rule. It has a preceding vowel, and if adding a ‘-ing’ or ‘-ed’ suffix it would also be followed by an ‘i’ or ‘e’ respectively. Without the ‘k’, the ‘c’ would thus end up pronounced as an /s/ instead of the /k/ that it is in the base form. Adding a ‘k’ isolates the ‘c’ from the following vowel, preserving the hard /k/ pronunciation.
Compare ‘sync’ and ‘syncing’, which does not add a ‘k’ because the ‘n’ proceeding the ‘c’ isolates it from the previous vowel.
There are a handful of exceptions to this, but they are exceedingly rare both in terms of how many there are and in terms of how frequently they are used in general conversation (the only ones I know of off the top of my head are ‘tic’ and ‘sic’, both of which double the ‘c’ instead of adding a ‘k’, neither is commonly seen in the present participle or past tense).
The rule mentioned at the top of the answer is also why some English verbs that end in a vowel followed by a consonant double the consonant in the present participle and past tense. The number of (orthographic) consonants following a vowel in written English impacts how that vowel is pronounced, and a doubled consonant is needed in those words to indicate the correct pronunciation.
To complement the other answers, I think it's worth remembering that, historically, language is always spoken before it's written (although this is less true since the advent of the internet). So the -ing rule is based on phonetics: you take the pronunciation of the word and add an "ing" sound at the end. Only after the word has already entered the vocabulary do people have to decide how to spell it. And sometimes they have made adjustments to better translate the pronunciation of the word to written form, like adding a 'k' to "mimicking". Another example is how the continuous form of "lie" is spelled "lying" rather than "lieing". "Lying" conveys better the pronunciation of the word to someone who is used to how English words are written (it rhymes with "crying" and "flying").