The reason why… that = We use the reason why before a clause
The reason why…because = Ungrammatical; do not use why and because in a single sentence.
The reason why is accompanied by the ‘result’ of the situation:
The reason why we couldn’t attend Barbara’s party was that it was raining heavily. √
The reason why he didn’t take her out to dinner was because he had got his pocket picked. ✗
The reason why he left the light on was that he had forgotten to switch it off. √
We sometimes omit why, especially in statements:
The reason (why) the doctor didn’t operate on him was that he was breathing his last.
Many established writers object to the use of the reason why as redundant. They argue why denotes ‘for what reason,’ exemplifying ‘reason for what reason.’ The counter-argument about it is that the redundancy is idiomatic and sometimes is ignored for the sake of clarity or emphasis.
Anyways, those who are averse to ‘the reason why’ will go for ‘the reason that’:
The reason (that) we didn’t leave the gate open is that the wind was blowing strongly. √
The reason … is because = The reason I didn’t attend the meeting is because I got caught in a traffic jam, is also not acceptable to many, on the grounds that either ‘the reason’ or ‘because’ is redundant; however, to avoid controversy, it is on the safe side to use ‘that’ instead (the reason I didn’t attend the meeting is that …) or rephrase altogether (I didn’t attend the meeting because …). All the same, both the constructions are well established and are acceptable in Standard English.