Mobile app: how to show the OTP my app sends in a popup without leaving my app?

Here is step by step description to achieve your requirement

1. Declare receiver in AndroidManifest

   <receiver android:name=".IncomingSms">   
    <intent-filter>
         <action android:name="android.provider.Telephony.SMS_RECEIVED" />
     </intent-filter>
   </receiver>
   

   2 Give read SMS permission in AndroidManifest

    <uses-permission android:name="android.permission.RECEIVE_SMS"></uses-permission>
    <uses-permission android:name="android.permission.READ_SMS" />
   

Complete code for AndroidManifest.xml File :

<?xml version="1.0" encoding="utf-8"?>
<manifest xmlns:android="http://schemas.android.com/apk/res/android"
    package="com.androidexample.broadcastreceiver"
    android:versionCode="1"
    android:versionName="1.0" >

    <application
        android:allowBackup="true"
        android:icon="@drawable/ic_launcher"
        android:label="@string/app_name"
        android:theme="@style/AppTheme" >
        <activity
            android:name="com.androidexample.broadcastreceiver.BroadcastNewSms"
            android:label="@string/app_name" >
            <intent-filter>
                <action android:name="android.intent.action.MAIN" />

                <category android:name="android.intent.category.LAUNCHER" />
            </intent-filter>
        </activity>

        <receiver android:name="com.androidexample.broadcastreceiver.IncomingSms">   
            <intent-filter>
                <action android:name="android.provider.Telephony.SMS_RECEIVED" />
            </intent-filter>
        </receiver>

    </application>
    <uses-sdk
        android:minSdkVersion="8"
        android:targetSdkVersion="17" />

    <uses-permission android:name="android.permission.RECEIVE_SMS"></uses-permission>
    <uses-permission android:name="android.permission.READ_SMS" />


</manifest>

Complete code for IncomingSms.java file :

public class IncomingSms extends BroadcastReceiver {

    // Get the object of SmsManager
    final SmsManager sms = SmsManager.getDefault();

    public void onReceive(Context context, Intent intent) {

        // Retrieves a map of extended data from the intent.
        final Bundle bundle = intent.getExtras();

        try {

            if (bundle != null) {

                final Object[] pdusObj = (Object[]) bundle.get("pdus");

                for (int i = 0; i < pdusObj.length; i++) {

                    SmsMessage currentMessage = SmsMessage.createFromPdu((byte[]) pdusObj[i]);
                    String phoneNumber = currentMessage.getDisplayOriginatingAddress();

                    String senderNum = phoneNumber;
                    String message = currentMessage.getDisplayMessageBody();

                    Log.i("SmsReceiver", "senderNum: "+ senderNum + "; message: " + message);


                   // Show Alert
                    int duration = Toast.LENGTH_LONG;
                    Toast toast = Toast.makeText(context, 
                                 "senderNum: "+ senderNum + ", message: " + message, duration);
                    toast.show();

                } // end for loop
              } // bundle is null

        } catch (Exception e) {
            Log.e("SmsReceiver", "Exception smsReceiver" +e);

        }
    }    
}

register above broadcast receiver in your activity and you will get your output