Why Do Gmail Sent Items Show in the Inbox?
Gmail makes use of a conversation view that puts your replies alongside responses from other contacts in the inbox. If you don't like this behavior, you can switch it off. Alternatively, you may have moved sent items into the inbox either automatically or by setting up a filter within Gmail.
1 Conversation View
The conversation view used by Gmail by default keeps emails from the same thread together as one entry in your inbox — this means messages between the same contacts with the same subject header are grouped in one list for easy reference. If you've just replied to an email thread, your response is shown in the inbox as the latest entry in the series. The feature makes it easier to keep track of ongoing conversations, but you can disable it through Gmail's settings if you wish.
2 Disabling Conversation View
Click the black cog icon at the top of the messages list in Gmail and choose "Settings" to disable the conversation view. The relevant option can be toggled on or off using the radio buttons by the Conversation View heading. With the conversation view turned off, messages appear separately in chronological order, with the newest at the top, and your own sent emails are excluded from the inbox. What's more, messages can be assigned separate labels with conversation view disabled.
3 Filters and Folders
Conversation view isn't the only reason why you might see sent items in the inbox. When viewing other folders there's a "Move to Inbox" button that you may have inadvertently pressed with sent messages selected. It's also worth checking your current Gmail filters by opening up the Settings page and switching to the Filters tab -- it's possible that one of the automatic rules listed here is sending some or all of your sent emails to the inbox based on certain criteria. Select a filter and click "Delete" if you want to remove it.
4 Email Clients
If you are accessing Gmail through a mobile or desktop client, such as Microsoft Outlook or Mail on Apple iOS, there are a number of reasons why you might see sent items in the inbox. Make sure you don't have any rules or filters set up that automatically move sent items into the inbox. If you're accessing Gmail using the IMAP protocol (see the app's settings for details), check that the Sent folder in Gmail is correctly mapped to the Sent folder in your email client. For further details, check the help documentation supplied with your email program.