1. Explain it simply
竟然 is an adverb meaning 'unexpectedly' or 'to one's surprise'. It marks a fact as hard to believe.
Word order: Subject + 竟然 + verb phrase. 竟然 always comes AFTER the subject and BEFORE the verb.
Use it when the result is the opposite of what you expected: 他竟然来了 means 'he actually came'.
竟然 comments on a real fact, not a guess - the thing really happened, it is just surprising.
2. Show examples
他竟然忘了我的名字。
Tā jìngrán wàng le wǒ de míngzi.
He actually forgot my name.
竟然 sits before the verb 忘
都八月了，天气竟然还这么冷。
Dōu bā yuè le, tiānqì jìngrán hái zhème lěng.
It is already August, yet the weather is unexpectedly still this cold.
surprise that it is still cold
他学中文才半年，竟然能看懂报纸了。
Tā xué zhōngwén cái bàn nián, jìngrán néng kàn dǒng bàozhǐ le.
He only studied Chinese for half a year, yet he can surprisingly read newspapers now.
unexpected ability after only half a year
3. Share a tip
Put 竟然 after the subject, right before the verb - say 他竟然来了, not 竟然他来了.