What is in the JSON payload Cloudwatch sends to SNS? How can I read that data?

The simplest and the most accurate way is to print(event) and actually see what's inside the payload, the official AWS doco provides the following structure for SNS event Amazon SNS Sample Event, reference

{
"Records": [
{
  "EventVersion": "1.0",
  "EventSubscriptionArn": eventsubscriptionarn,
  "EventSource": "aws:sns",
  "Sns": {
    "SignatureVersion": "1",
    "Timestamp": "1970-01-01T00:00:00.000Z",
    "Signature": "EXAMPLE",
    "SigningCertUrl": "EXAMPLE",
    "MessageId": "95df01b4-ee98-5cb9-9903-4c221d41eb5e",
    "Message": "Hello from SNS!",
    "MessageAttributes": {
      "Test": {
        "Type": "String",
        "Value": "TestString"
      },
      "TestBinary": {
        "Type": "Binary",
        "Value": "TestBinary"
      }
    },
    "Type": "Notification",
    "UnsubscribeUrl": "EXAMPLE",
    "TopicArn": topicarn,
    "Subject": "TestInvoke"
  }
}
]
}

also, this CloudWatch specific example could be useful

{
"Records": [
{
  "EventSource": "aws:sns",
  "EventVersion": "1.0",
  "EventSubscriptionArn": "arn:aws:sns:eu-west-1:000000000000:cloudwatch-alarms:xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx",
  "Sns": {
    "Type": "Notification",
    "MessageId": "xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx",
    "TopicArn": "arn:aws:sns:eu-west-1:000000000000:cloudwatch-alarms",
    "Subject": "ALARM: \"Example alarm name\" in EU - Ireland",
    "Message": "{\"AlarmName\":\"Example alarm name\",\"AlarmDescription\":\"Example alarm description.\",\"AWSAccountId\":\"000000000000\",\"NewStateValue\":\"ALARM\",\"NewStateReason\":\"Threshold Crossed: 1 datapoint (10.0) was greater than or equal to the threshold (1.0).\",\"StateChangeTime\":\"2017-01-12T16:30:42.236+0000\",\"Region\":\"EU - Ireland\",\"OldStateValue\":\"OK\",\"Trigger\":{\"MetricName\":\"DeliveryErrors\",\"Namespace\":\"ExampleNamespace\",\"Statistic\":\"SUM\",\"Unit\":null,\"Dimensions\":[],\"Period\":300,\"EvaluationPeriods\":1,\"ComparisonOperator\":\"GreaterThanOrEqualToThreshold\",\"Threshold\":1.0}}",
    "Timestamp": "2017-01-12T16:30:42.318Z",
    "SignatureVersion": "1",
    "Signature": "Cg==",
    "SigningCertUrl": "https://sns.eu-west-1.amazonaws.com/SimpleNotificationService-xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx.pem",
    "UnsubscribeUrl": "https://sns.eu-west-1.amazonaws.com/?Action=Unsubscribe&SubscriptionArn=arn:aws:sns:eu-west-1:000000000000:cloudwatch-alarms:xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx",
    "MessageAttributes": {}
  }
}
]
}

The actual payload sent by Cloudwatch to SNS,

{
  "AlarmName": "Example alarm name",
  "AlarmDescription": "Example alarm description.",
  "AWSAccountId": "000000000000",
  "NewStateValue": "ALARM",
  "NewStateReason": "Threshold Crossed: 1 datapoint (10.0) was greater than or equal to the threshold (1.0).",
  "StateChangeTime": "2017-01-12T16:30:42.236+0000",
  "Region": "EU - Ireland",
  "OldStateValue": "OK",
  "Trigger": {
    "MetricName": "DeliveryErrors",
    "Namespace": "ExampleNamespace",
    "Statistic": "SUM",
    "Unit": null,
    "Dimensions": [],
    "Period": 300,
    "EvaluationPeriods": 1,
    "ComparisonOperator": "GreaterThanOrEqualToThreshold",
    "Threshold": 1
  }
}

If you want to read this event in Lambda,

import json

def lambda_handler(event, context):
    alert_message = json.loads(json.dumps(event))["Records"][0]["Sns"]["Message"]

    return {
        'statusCode': 200,
        'body': json.dumps(alert_message)
    }