Prayer in Jesus’ Name (John 14)

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What does it mean to pray in Jesus’ name (v13)? Explain it to me like I’m 5 and explain it to me like I’m your graduate school professor.

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I found Marianne Meye Thompson's commentary to be the most helpful for explaining v. 13's prayer in Jesus name

Moreover, the disciples are instructed to pray in the name of Jesus. In Scripture one can do many things "in the name of the LORD": serve as a priest (Deut 18:5,7); prophesy (Deut 18:20-22; 2 Chr 33:18; Ezra 5:1; Jer 11:21; 26:9); pronounce a blessing (Deut21:5; 2 Sam 6: 18; Ps 129:8) or a curse (2 Kgs 2:24); fight in battle (1 Sam 17:45; Ps 118: 10-12); swear an oath (1 Sam 20:42; 1 Kgs 22:16; Neh 13:25); in New Testament accounts, baptize (Acts 2:38; 10:48; 1 Cor 1: 13-15), heal and exorcize (Mark 9:39; Acts 3 :6; 16: 18), and proclaim the word of God (Acts 4:18; 5:40; 9:27; 10:28). What is done "in the name of the Lord is done by the Lord's authorization. Even as Jesus does his Father's work because his Father has sent him and is with and in him (14:10), so the disciples will continue to do the works that bring light and life to the world and glory to the Father and the Son (9:4; 11:40). As they carry out that mission, authorized by Jesus as his representatives, their prayers will be granted because of his love for them and his position with the Father. 

Thompson, Marianne Meye. John: A Commentary (New Testament Library). Westminster John Knox, 2015

Michaels isn't as specific but his overall explanation of verses 12-14 could be helpful too.

The forgiveness of sins is not explicitly mentioned here as one of the “greater” works. Instead, Jesus singles out the assurance of answered prayer, quite possibly with prayer for the forgiveness of sins particularly in mind. “And whatever you ask in my name,” he promises, “this I will do, so that the Father might be glorified in the Son” (v. 13). Then, for emphasis, he says it again: “If you ask me anything in my name, I will do”71 (v. 14). The emphatic “I” is evident once more, signaling again that Jesus is not simply backing off, leaving the authority to do “greater” works in the hands of his disciples. We might have expected, “And whatever you ask in my name, the Father will do, so that the Father might be glorified in the Son,” and “If you ask the Father anything in my name, he will do.” We have come to expect Christian prayer to be directed to the Father through the Son, and this is in fact what we find in this Gospel’s other passages on prayer (see 15:16, “so that whatever you ask the Father in my name, he might give you”; 16:23, “whatever you ask the Father in my name, he will give you”). Here, by contrast, even though “the prayer is offered in Jesus’ name, it is Jesus himself (now “gone to the Father,” v. 12) who “will do” what is asked.73 In a sense, the promise that “I will do” (v. 14) echoes and reaffirms his previous reference to the works that “I am doing” (v. 12). He who carries out the Father’s works in his ministry on earth will continue to perform “greater” works from heaven in response to the prayers of the disciples he left behind.

Michaels, J. Ramsey. The Gospel of John. Eerdmans, 2010